Part Of: Machine Learning sequence
Content Summary: 800 words, 8 min read

Parameter Space vs Feature Space

Let’s recall the equation of a line.

$y = mx + b$ where $m$ is the slope, and $b$ is the y-intercept.

The equation of a line is a function that maps from inputs (x) to outputs (y). Internal to that model (the knobs inside the box) reside parameters like $m$ and $b$ that mold how the function works.

Any model $k$ can be uniquely described by its parameters $\{ m_k, b_k \}$. Just as we can plot data in a data space using Cartesian coordinates $(x, y)$, we can plot models in a parameter space using coordinates $(m,b)$.

As we traverse parameter space, we can view the corresponding models in data space.

As we proceed, it is very important to hold these concepts in mind.

Loss Functions

Consider the following two regression models. Which one is better?

The answer comes easily: Model A. But as the data volume increases, choosing between models can become quite difficult. Is there a way to automate such comparisons?

Regression models are functions of the form $\hat{y} = f(\textbf{x})$ where x is the vector of features (predictors) used to generate the label (prediction). We can define error as $\hat{y} - y$. In fact, we typically reserve the word error for test data, and residual for train data. Here are the residuals for our two regression models:

The larger the residuals, the worse the model. Let’s use the residual vector to define a loss function. To do this, in the language of database theory, we need to aggregate the column down to a scalar. In the language of linear algebra, we need to compute the length of the vector.

Everyone agrees that residuals matter, when deriving the loss function. Not everyone agrees how to translate the residual vector into a single number. Let me add a couple examples to motivate:

Sum together all prediction errors.

But then an deeply flawed model with residual vector  $[ -30, 30, -30, 30]$ earns the same score as a “perfect model” $[0, 0, 0, 0]$. The morale: positive and negative errors should not cancel each other out.

Sum together the magnitude of the prediction errors.

But then a larger dataset costs more than a small one. A good model against a large dataset with residual vector $[ 1, -1, 1, -1, 1, -1 ]$ earns the same score as a poor model against small data $[ 3, -3 ]$. The morale: cost functions should be invariant to data volume.

Find the average magnitude of the prediction errors.

This loss function suffers from fewer bugs. It even has a name: Mean Absolute Error (MAE), also known as the L1-norm.

There are many other valid ways of defining a loss function that we will explore later. I  just used the L1-norm to motivate the topic.

Let’s return to the question of evaluating model performance. For the following five models, we intuitively judge their performance as steadily worsening:

With loss functions, we convert these intuitive judgments into numbers. Let’s include these loss numbers in label space, and encode them as color.

Still with me? Something important is going on here.

We have examined the loss of five models. What happens if we evaluate two hundred different models? One thousand? A million? With enough samples, we can gain a high-resolution view of the loss surface. This loss surface can be expressed with loss as color, or loss as height along the z-axis.

In this case, the loss surface is convex: it is in the shape of a bowl.

Navigating the Loss Surface

The notion of a loss surface takes a while to digest. But it is worth the effort. The loss surface is the reason machine learning is possible.

By looking at a loss surface, you can visually identify the global minimum: the model instantiation with the least amount of loss. In our example above, that is $(7,-14)$ encodes the model $y = 7x - 14$ with the smallest loss $L(\theta) = 2$.

Unfortunately, computing the loss surface is computationally intractable. It takes too long to calculate the loss of every possible model. How can we do better?

2. Figure out how to improve it.
3. Repeat.

One useful metaphor for this kind of algorithm is a flashlight in the dark. We can’t see the entire landscape, but our flashlight provides information about our immediate surroundings.

But what local information can we use to decide where to move in parameter space? Simple: the gradient (i.e., the slope)! If we move downhill in this bowl-life surface, we will come to a rest at the set of best parameters.

A ball rolling down a hill.

This is how gradient descent works, in both spaces:

This is how prediction machines learn from data.

Until next time.

# Bias vs Variance

Part Of: Machine Learning sequence
Followup To: Regression vs Classification
Content Summary: 800 words, 8 min read

A Taxonomy of Models

Last time, we discussed how to create prediction machines. For example, given an animal’s height and weight, we might build a prediction machine to guess what kind of animal it is. While this sounds complicated, in this case prediction machines are simple region-color maps, like these:

These two classification models are both fairly accurate, but differ in their complexity.

But it’s important to acknowledge the possibility of erroneous-simple and erroneous-complex models. I like to think of models in terms of an accuracy-complexity quadrant.

This quadrant is not limited to classification. Regression models can also vary in their accuracy, and their complexity.

A couple brief caveats before we proceed.

• This quadrant concept is best understood as a two-dimensional continuum, rather than a four-category space. More on this later.
• Here “accuracy” tries to capture lay intuitions about prediction quality & performance. I’m not using it in the metric sense of “alternative to F1 score”.

Formalizing Complexity

Neural networks are often used as classification models against large numbers of images. The complexity of the models tends to correlate with the number of layers. For some models then, complexity is captured in the number of parameters.

While not used much in the industry, polynomial models are pedagogically useful examples of regression models. Here, the degree of the polynomial expresses the complexity of the model: a degree-eight polynomial has more “bumps” than a degree-two polynomial.

Consider, however, the difference between the following regression models

$y_A = 4x^4 + 0.0001x^3 + 0.0007x^2 + 2.1x + 7$
$y_B = 4x^4 + 2.1x + 7$

Model A uses five parameters; Model B uses three. But their predictions are, for all practical purposes, identical. Thus, the size of each parameter is also relevant to the question of complexity.

The above approaches rely on the model’s parameters (its “visceral organs”) to define complexity. But it is also possible to rely on the model’s outputs (its “behaviors”) to achieve the same task. Consider again the classification decision boundaries above. We can simply measure the spatial frequency (the “squiggliness” of the boundary) as another proxy towards complexity.

Here, then, are three possible criteria for complexity:

1. Number of parameters
2. Size of parameters
3. Spatial frequency of decision manifold

Thus, operationalizing the definition of “complexity” is surprisingly challenging. But there is another way to detect whether a model is too complex…

Simplicity as Generalizability

Recall our distinction between training and prediction:

We compute model performance on historical data. We can contrast this with model performance again future data.

Take a moment to digest this image. What is it telling you?

Model complexity is not merely aesthetically ugly. Rather, complexity is the enemy of generalization. Want to future-proof your model? Simplicity might help!

Underfitting vs Overfitting

There is another way of interpreting this tradeoff, that emphasizes the continuity of model complexity. Starting from a very simple model, increases in model complexity will improve both historical and future error. The best response to underfitting is increasing the expressivity of your model.

But at a certain point, your model will become too complex, and begin to overfit the data. At that point, your historical error will continue to decrease, but your future error will increase.

Data Partitioning: creating a Holdout Set

So you now appreciate the importance of striking a balance between accuracy and simplicity. That’s all very nice conceptually, but how might you go about building a well-balanced prediction machine?

The bias trade-off is only apparent when the machine is given new data!  “If only I had practiced against unseen test data earlier”, the statistician might say, “then I could have discovered how complex to make my model before it was too late”.

Read the above regret again. It is the germinating seed of a truly enormous idea.

Many decades ago, some creative mind took the above regret and sought to reform it: “What stops me from treating some of my old, pre-processed data as if it were new? Can I not hide data from myself?”

This approach, known as data partitioning, is now ubiquitous in the machine learning community.  Historical-Known data is the training set, Historical-Novel data is the test set, aka the holdout set.

How much data should we put in the holdout set? While the correct answer ultimately derives from the particular application domain, a typical rule of thumb:

• On small data (~100 thousand records), data are typically split to 80% train, 20% test
• On large data (~10 billion records), data are typically split to 95% train, 5% test

Next time, we will explore cross-validation (CV). Cross-validation is sometimes used instead of, and other times in addition to, data partitioning.

See you then!

# Counting: The Fourfold Way

Part Of: Statistics, Algebra sequences
Content Summary: 1100 words, 11 min read

The Fundamental Principle of Counting

We often care to count the number of possible outcomes for multiple events.

Example 1. Consider purchasing a lunch with the following components:

• Burger $b \in \{ Chicken, Beef \}$
• Side $s \in \{ Fries, Chips \}$
• Drink $d \in \{ Fanta, Coke, Sprite \}$

How many lunch outcomes are possible?

Three approaches to counting suggest themselves. We might make a list. But this process can be error prone. Other representations are more systematic: we can build a tree, or imagine a (hyper)-volume. Each strategy converges on the same answer: 12 possible lunches.

Can we generalize? Yes, with the help of the fundamental principle of counting (aka the rule of multiplication). For any event $A$ with $a$ possible outcomes, and another event $B$ with $b$ possible outcomes, the number of possible outcomes for composite event $A \cup B$ is $a*b$

How does counting work for a repeating event? For an event with $n$ possibilities occurs $k$ times, there are $k^n$ possible outcomes.

Example 2. How many numbers can be represented by a byte (8 bits)?

Each bit has two possible assignments: zero or one. For eight such “bit events”, we have $2^8 = 256$ possible outcomes.

Permutations

Example 3. A trifecta bet guesses which horse will place first, which second, and which third.  How many such bets are possible in a 9-horse race?

Each medal has nine possible assignments. For three such “medal events”, we have $9^3 = 729$ possible outcomes.

This answer is completely wrong. To understand why, consider the lottery machine:

• In Example 2, a single value (e.g., 0) can freely be assigned to multiple bits. Every time you draw a bit from the “possibility machine”, it is replaced when the next bit is drawn. Sampling with replacement means that each event is exactly the same.
• In Example 3, a single horse (e.g., Secretariat) cannot be assigned multiple medals. Every time you draw a horse from the “possibility machine”, it cannot be drawn for subsequent events. Sampling without replacement means that each event has diminishing numbers of possibilities.

Definition 4. A permutation is a list of outcomes drawn without replacement.

For the trifecta bet, how many permutations exist? Well, 9 different horses that earn the gold. Given that one horse won the gold, 8 different horses that can earn the silver. Then there are 7 different horses that can earn bronze. Thus, there are $9 \times 8 \times 7 = 504$ possible trifecta bets.

Similar to how exponentiation is defined as repeated multiplication, a factorial is defined as slowly-decrementing multiplication.

$9! = 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1$

But we only want $9 \times 8 \times 7$. How can we get rid of the other terms? By division, of course!

$9 \times 8 \times 7 = \dfrac{9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}{6 \times 5 \times 4 \times 3 \times 2 \times 1} = \dfrac{9!}{6!}$

Why did we use the number 6? Because if three of our nine horses place, six do not place. So a more general way to write this equation,

$\dfrac{9!}{9-3!}$

More generally, if you have n items and want to find the number of ways k items can be ordered:

Equation 5: Permutation. $P(n, k) = \dfrac{n!}{(n-k)!}$

Combinations

In contrast to permutations, for combinations, order doesn’t matter. A permutation is a list, a combination is a set.

A boxed trifecta bet requires correctly which three horses will place first, second and third (order doesn’t matter). A trifecta bet selects a permutations; a boxed trifecta bet selects a combination.

Imagine only four horses in the race. That’s $P(4,3) = \dfrac{4!}{4-3!} = 24$ possible trifecta bets. But how many boxed trifecta bets are possible?

Combinations treat duplicates as a single entry. For example, $abc$ and $acb$ are equivalent for a boxed trifecta bet. We can identify four groups, with six equivalent permutations each:

In general, how many winner duplicates exist? How many ways can we shuffle $k$ winners? Well, if you have k winners and are wondering how many permutations exist for that entire set… that’s $P(k,k)$!

Equation 6: Combination. $C(n, k) = \dfrac{P(n,k)}{P(k,k)} = \dfrac{P(n,k)}{k!} = \dfrac{n!}{(n-k)! k!}$

For an example of combinations used to solve a real problem, I recommend this post.

The Fourth Way: Stars and Bars

Example 7. You have $k=3$ cookies $a, b, c, d$ to give to $n=4$ kids. How many possible ways are there to do so?

In the case of medals and horses, we claimed four solutions: $\{ a, b, c\}$$\{ a, b, d\}$$\{ a, c, d\}$, and $\{ b, c, d\}$. But there is an important difference: horse-less medals are impossible, but cookie-less children are not! So we need to account for situations like $\{ a, a, a\}$, with one child getting all of the cookies.

We can use the traditional bins-as-containers metaphor to visualize outcomes (top row). Or we can instead visualize bin boundaries (bottom row). This visualization strategy is called stars and bars.

How many kid-cookie outcomes are possible? The answer becomes apparent only if we use stars and bars  (bottom row). Every possible shuffling of the stars in those squares produces a valid event. That is, $\binom{6}{3}$.

How many objects are possible in general? There are $n$ stars (kids). Since bars represent bin boundaries, there are $n-1$ bars. Thus:

Equation 8: Multi-Combination. $C(n, k) = \binom{n+(k-1)}{k}$

The Fourfold Way

Every example we have seen differentiates possibilities and outcomes. We will use the metaphor of balls for outcomes (something concrete) and bins for possibilities (something to “clothe” outcomes).

Equation 9. An event is a function that maps outcomes to possibilities:

$Event : Outcomes \rightarrow Possibilities$

This function can be compactly represented as $bbd$.

Functions require every element of the domain to map to the codomain. Event functions require no unrealized outcomes. That is: every outcomes manifests a possibility. Every ball is given a bin.

We saw previously that combinations and permutations don’t allow events like $aab$ and $ccc$. A single horse cannot win multiple medals. Multiply-realized possibilities are not allowed.

Recall the definitions of injective, surjective and bijective functions. This requirement is the injective property. Sampling without replacement is the same thing as injectivity.

Tuple, permutation, combination, and multi-combination. This is the fourfold way.  The Way can be made more general by counting situations where the possibilities are unlabeled, and the event function meets the surjection property. But for details on the more complete twelvefold way, I recommend this post.

Towards a Rosetta Stone

Now, consider all possible functions for 4 bins and 3 balls:

What do the equations above have to do with this shape? Well, each way of counting corresponds with a different subset of this broader shape:

I leave it to the interested reader to ponder, how such a jagged shape can be represented by these four relatively clean formulae.

Until next time.

Related Resources

https://www.ece.utah.edu/eceCTools/Probability/Combinatorics/ProbCombEx15.pdf

https://wizardofodds.com/games/poker/

http://www.math.hawaii.edu/~ramsey/Probability/PokerHands.html

# The Symmetric Group

Part Of: Algebra sequence
Followup To
Content Summary: 1800 words, 18 min read

On Permutations

In the last few posts, we have discussed algebraic structures whose sets contain objects (e.g., numbers). Now, let’s consider structures over a set of functions, whose binary operation is function composition.

Definition 1. Consider two functions $f$ and $g$. We will denote function composition of $g(f(x))$ as $f \bullet g$. We will use this notation instead of the more common $g \circ f$.  Both represent the idea “apply $f$, then $g$“.

Consider $G = (\left\{ f(x) = 2x, g(x) = x+1, h(x) = 2x + 2, i(x) = 2x+1 \right\}, \bullet)$

Is this a group? Let’s check closure:

• $g \bullet f = 2(x+1) = h \in G$
• $f \bullet g = (2x)+1 = i \in G$
• $f \bullet h = 2(2x)+2 \not\in G$

Closure is violated. $G$ isn’t even a magma! Adding $j(x) = 4x+2$ to the underlying set exacerbates the problem: then both $f \bullet j \not\in G$ and $g \bullet j \not\in G$.

So it is hard to establish closure under function composition. Can it be done?

Yes. Composition exhibits closure on sets of permutation functions. Recall that a permutation is simply a bijection: it re-arrange a collection of things. For example, here are the six possible bijections over a set of three elements.

Definition 2. The symmetric group $S_n$ denotes composition over a set of all bijections (permutations) over some set of $n$ objects. The symmetric group is then of order $n!$.

The underlying set of $S_{3}$ is the set of all permutations over a 3-element set. It is of order $3! = 3 \times 2 \times 1 = 6$.

This graphical representation of permutations is rather unwieldy. Let’s switch to a different notation system!

Notation 3: Two Line Notation. We can use two lines to denote each permutation mapping $\phi$. The top row represents the original elements $x$, the bottom represents where each element has been relocated $\phi(x)$.

Two line notation is sometimes represented as an array, with the top row as matrix row, and bottom denoting matrix column. Then the identity matrix represents the identity permutation.

Definition 4. A cycle is a sequence of morphisms that forms a closed loop. An n-cycle is a cycle of length n. A 1-cycle does nothing. A 2-cycle is given the special name transposition. $S_3$ has two permutations with 3-cycles: can you find them?

Theorem 5. Cycle Decomposition Theorem. Every permutation can be decomposed into disjoint cycles. Put differently, a node cannot participate in more than one cycle. If it did, its parent cycles would merge.

Notation 6: Cycle Notation. Since permutations always decompose into cycles, we can represent them as $(A_1\ A_2\ \ldots\ A_n)$, pronounced “$A_1$ goes to $A_2$ goes to …”.

Cycle starting element does not matter: $(A\ C\ B) = (C\ B\ A) = (B\ A\ C)$.

The Cycle Algorithm

It is difficult to tell visually the outcome of permutation composition. Let’s design an algorithm to do it for us!

Algorithm 7: Cycle Algorithm. To compose two permutation functions $a(S)$ and $b(S)$, take each element $s \in S$ and follow its arrows until you find the set of disjoint cycles. More formally, compose these functions $x$ times until you get $(a \bullet b)^x(s) = s$.

Here’s a simple example from $S_{3} = ( \left\{ 0, 1, 2, 3, 4, 5 \right\}, \bullet )$.

Make sense? Good! Let’s try a more complicated example from $S_4$.

A couple observations are in order.

• 1-cycles (e.g., $(A)$) can be omitted: their inclusion does not affect algorithm results.
• Disjoint cycles commute: $(B\ D),(A\ C) = (A\ C),(B\ D)$. Contrast this with composition, which does not commute $(B\ D) \bullet (A\ C) \neq (A\ C) \bullet (B\ D)$.

Now, let’s return to $S_{3}$ for a moment, with its set of six permutation functions. Is this group closed? We can just check every possible composition:

From the Cayley table on the right, we see immediately that $S_{3}$ is closed (no new colors) and non-Abelian (not diagonal-symmetric).

But there is something much more interesting in this table. You have seen it before. Remember the dihedral group $D_{3}$? It is isomorphic to $S_{3}$!

If you go back to the original permutation pictures, this begins to make sense. Permutations $1$ and $2$ resemble rotations/cycles; $3$, $4$, and $5$ perform reflections/flips.

Generators & Presentations

In Theorem 7, we learned that permutations decompose into cycles. Let’s dig deeper into this idea.

Theorem 8. Every n-cycle can be decomposed into some combination of 2-cycles. In other words, cycles are built from transpositions.

The group $S_{3} = \left\{ 0, 1, 2, 3, 4, 5 \right\}, \bullet)$ has three transpositions $3 = (A\ B)$, $4 = (B\ C)$, and $5 = (A\ C)$.

Transpositions are important because they are generators: every permutation can be generated by them. For example, $1 = 3 \bullet 4$. In fact, we can lay claim to an even stronger fact:

Theorem 9. Every permutation can be generated by adjacent transpositions. Every permutation $S_{n} = \langle (1 2), (2 3), \ldots, (n-1\ n) \rangle$.

By the isomorphism $S_3 \cong D_3$ , we can generate our “dihedral-looking” Cayley graph by selecting generators $1=(A\ B\ C)$ and $3 = (B\ C)$.

But we can use Theorem 9 to produce another, equally valid Cayley diagram. There are two adjacent transpositions in $S_{3}$: $3$ and $4$. All other permutations can be written in terms of these two generators:

• $0 = 3 \bullet 3$.
• $1 = 3 \bullet 4$.
• $2 = 4 \bullet 3$.
• $5 = 3 \bullet 4 \bullet 3$

This allows us to generate a transposition-based Cayley diagram. Here are the dihedral and transposition Cayley diagrams, side by side:

We can confirm the validity of the transposition diagram by returning to our multiplication table: $1 \bullet 3 = 5$ means a green arrow $1 \rightarrow 5$.

Note that the transposition diagram is not equivalent cyclic group $C_6$, because arrows in the latter are monochrome and unidirectional.

We’re not quite done! We can also rename our set elements to employ generator-dependent names, by “moving clockwise”:

We could just as easily have “moved counterclockwise”, with names like $3 \mapsto r$, $1 \mapsto rl$. And we can confirm by inspection that, in fact, $rl = lrlr$ etc.

Using the original clockwise notation, one presentation of $S_3$ becomes:

$S_3 = \langle l, r \mid r^2 = e, l^2 = e, (lr)^3 = e \rangle$

Towards Alternating Groups

Any given permutation can be written as a product of permutation. Consider, for example, the above equalities

• $1 = rl = lrlr = (lr)^3rl$. These have 2, 4, and 8 permutations, respectively.
• $5 = lrl = lrl^3 = l^3r^3l^3$. These have 3, 5, and 9 permutations, respectively.

Did you notice any patterns in the above lists? All expressions for $1$ require an even number of transpositions, and all expressions of $5$ require an odd number. In other words, the parity (evenness or oddness) of a given permutation doesn’t seem to be changing. In fact, this observation generalizes:

Theorem 10. For any given permutation, the parity of its transpositions is unique.

Thus, we can classify permutations by their parity. Let’s do this for $S_3$:

• $0, 1, 2$ are even permutations.
• $3, 4, 5$ of odd permutations.

Theorem 11. Exactly half of $S_n$ are even permutations, and they form a group called the alternating group $A_n$. Just as $\lvert S_n \rvert = n!$, $A_n$ has $\dfrac{n!}{2}$ elements.

Why don’t odd permutations form a group? For one thing, it doesn’t contain the identity permutation, which is always even.

Let’s examine $A^3 = (\left\{ 0, 1, 2 \right\}, \bullet )$ in more detail. Does it remind you of anything?

It is isomorphic to the cyclic group $C_3$ ..!

We have so far identified the following isomorphisms: $S_3 \cong D_3$ and $A_3 \cong C_3$. Is it also true that e.g., $S_4 \cong D_4$ and $A_4 \cong C_4$?

No! Recall that the $\lvert D_n \rvert=2n$ and $\lvert C_n \rvert = n$Only $n \neq 3$, these sets are not even potentially isomorphic.  For example:

• $\lvert D_4 \rvert=2 \times 4 = 8 \neq 24 = 4! = \lvert S_4 \rvert$.
• $\lvert C_4 \rvert= 4 \neq 12 = \frac{24}{2} = \lvert A_4 \rvert$.

For these larger values of $n$, the symmetric group is much larger than dihedral and cyclic groups.

Applications

Why do symmetric & alternating groups matter? Let me give two answers.

Perhaps you have seen the quadratic equation, the generic solution to quadratic polynomials $ax^2 + bx + c = 0$.

$x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Analogous formulae exist for cubic (degree-3) and quartic (degree-4) polynomials. 18th century mathematics was consumed by the theory of equations: mathematicians attempting to solve quintic polynomials (degree-5). Ultimately, this quest proved to be misguided: there is no general solution to quintic polynomials.

Why should degree-5 polynomials admit no solution? As we will see when we get to Galois Theory, it has to do with the properties of symmetric group $S_5$.

A second reason to pay attention to symmetric groups comes from the classification theorem of finite groups. Mathematicians have spent decades exploring the entire universe of finite groups, finding arcane creatures such as the monster group, which may or may not explain features of quantum gravity.

One way to think about group space is by the following periodic table:

Image courtesy of Samuel Prime

Crucially, in this diverse landscape, the symmetric group plays a unique role:

Theorem 12: Cayley’s Theorem. Every finite group is a subgroup of the subgroup $S_n$, for some sufficiently large $n$.

For historical reasons, subgroups of the symmetric group are usually called permutation groups.

Until next time.

Wrapping Up

Takeaways:

• The symmetric group $S_n$ is set of all bijections (permutations) over some set of $n$ objects, closed under function composition.
• Permutations can be decomposed into disjoint cycles: cycle notation uses this fact to provide an algorithm to solve for arbitrary compositions.
• All permutations (and hence, the symmetric group) can be generated by adjacent transpositions. This allows us to construct a presentation of the symmetric group.
• Permutations have unique parity: thus we can classify permutations as even or odd. The group of even presentations is called the alternating group $A_n$.
• It can be shown that $S_3 \cong D_3$ and $A_3 cong C_3$. However, for larger $n$, the symmetric and alternating group are much larger than cyclic and dihedral groups.

The best way to learn math is through practice! If you want to internalize this material, I encourage you to work out for yourself the Cayley table & Cayley diagram for $S_4$. 🙂

Related Resources

For a more traditional approach to the subject, these Harvard lectures are a good resource.

# An Introduction to Geometric Group Theory

Part Of: Algebra sequence
Followup To: An Introduction to Abstract Algebra
Content Summary: 1500 words, 15 min read

Last time, we saw algebraic structures whose underlying sets were infinitely large (e.g., the real numbers $\mathbb{R}$). Are finite groups possible?

Consider the structure $( \left\{ 0, 1, 2, 3 \right\}, +)$. Is it a group? No, it isn’t even a magma: $2 + 3 \not\in \left\{ 0, 1, 2, 3 \right\}$! Is there a different operation that would produce closure?

Modular arithmetic is the mathematics of clocks. Clocks “loop around” after 12 hours. We can use modulo-4 arithmetic, or $+_{4}$, on $\left\{ 0, 1, 2, 3 \right\}$. For example, $2 +_{4} 3 = 1$.

To check for closure, we need to add all pairs of numbers together, and verify that each sum has not left the original set. This is possible with the help of a Cayley table. You may remember these as elementary school multiplication tables 😛 .

By inspecting this table, we can classify $Z_4 = ( \left\{ 0, 1, 2, 3 \right\} ), +_{4})$.

1. Does it have closure? Yes. Every element in the table is a member of the original set.
2. Does it have associativity? Yes. (This cannot be determined by the table alone, but is true on inspection).
3. Does it have identity? Yes. The rows and columns associated with 0 express all elements of the set.
4. Does it have inverse? Yes. The identity element appears in every row and every column.
5. Does it have commutativity? Yes. The table is symmetric about the diagonal.

Therefore, $Z_4$ is an abelian group.

An Example Using Roots of Unity

Definition 1. A group is said to be order $n$ if its underlying set has cardinality $n$.

So $Z_4$ is order 4. What other order-4 structures exist?

Consider the equation $i^4 = -1$. Its solutions, or roots, is the set $\left\{ 1, i, -1, -i \right\}$. This set is called the fourth roots of unity.

So what is the Cayley table of this set under multiplication $R_{4} = ( \left\{ 1, i, -1, -i \right\}, *)$? In the following table, recall that $i = \sqrt{-1}$, thus $i^2 = (sqrt{-1})^2 = -1$.

Something funny is going on. This table (and its colors) are patterned identically to $Z_4$! Recall that a binary operation is just a function $f : A \times A \rightarrow A$. Let’s compare the function maps of our two groups:

These two groups for structurally identical: two sides of the same coin. In other words, they are isomorphic, we write $Z_{4} \cong R_{4}$. Let us call this single structure $C_4$.

But why are these examples of modular arithmetic and complex numbers equivalent?

One answer involves an appeal to rotational symmetry. Modular arithmetic is the mathematics of clocks: the hands of the clock rotating around in a circle. Likewise, if the reals are a number line, complex numbers are most naturally viewed as rotation on a number plane.

This rotation interpretation is not an accident. It helps use more easily spot other instances of $C_4$. Consider, for instance, the following shape.

On this shape, the group of rotations that produce symmetry is $W_4 = (\left\{ 0, 90, 180, 270 \right\}, \text{rotate})$. Inspection reveals that this, too, is isomorphic to $C_{4}$!

Towards The Presentation Formalism

We describe $C_3$ as a cyclic group, for reasons that will become clear later.

Theorem 2. For every cyclic group $C_n$, there exists some generator $g$ in its underlying set such that every other set element can be constructed by that generator.

Definition 3. When a generator has been identified, we can express a group’s underlying set with generator-dependent names. Two notation are commonly used in practice:

1. In multiplicative notation, the elements are renamed $\left\{ e, r, r^2, r^3 \right\}$, where r is any generator.
2. Similarly, in additive notation, the elements become $\left\{ e, r, 2r, 3r \right\}$.

These two notation styles are interchangeable, and a matter of taste. In my experience, most mathematicians prefer multiplicative notation.

What generators exist in $C_4$? Let’s look at our three instantiations of this group:

• In modular arithmetic, you can recreate all numbers by $0 + 1 + 1 + \ldots$. But you can also recreate them by $0 + 3 + 3 + \ldots$.
• In complex numbers, you can visit all numbers by multiplying by $i$, or multiplying by $-i$. Only $-1$ fails to be a generator.
• In our rotation symmetry shape, two generators exist: clockwise $90 \circ$ rotation, and counterclockwise $90 \circ$ rotation.

For now, let’s rename all elements of $C_{4}$ to be $C_4 = (\left\{ 0, 1, 2, 3 \right\}, +) = \langle 1 \rangle = \langle 3 \rangle$.

Okay. But why is $2$ not a generator in $C_4$?

Theorem 4. For finite groups of order $n$, each generator must be coprime to $n$. That is, their greatest common divisor $\text{gcd}(g, n) = 1$.

• $2$ not a generator in $C_4$ because it is a divisor of $| \left\{ 0, 1, 2, 3 \right\} | = 4$.
• What are the generators in $C_5$? All non-identity elements: $C_{5} = \langle 1 \rangle = \langle 2 \rangle = \langle 3 \rangle = \langle 4 \rangle$.
• What are the generators in $C_6$? Only 1 and 5: $C_{5} = \langle 1 \rangle = \langle 5 \rangle$.

We just spent a lot of words discussing generators. But why do they matter?

Generators are useful because they allow us to discover the “essence” of a group. For example, the Rubik’s cube has $5.19 \times 10^{20}$ configurations. It would take a long time just writing down such a group. But it has only six generators (one for a $90 \circ$ rotation along each of its faces) which makes its presentation extremely simple.

Another way to think about it is, finding generators is a little bit like identifying a basis in linear algebra.

Towards Cayley Diagrams

Definition 5. We are used to specifying groups as set-operator pairs. A presentation is an generator-oriented way to specify the structure of a group. A relator is defined as constraints that apply to generators. A presentation is written $\langle \text{generators} \mid \text{relators} \rangle$

• In multiplicative notation: $C_4 = \langle r \mid r^3 = e \rangle$.
• In additive notation: $C_4 = \langle r \mid 3r = e \rangle$.

The $= e$ suffix is often left implicit from presentations (e.g., $C_4 = \langle r \mid r^n \rangle$) for the sake of concision.

Definition 6. A Cayley diagram is used to visualize the structure specified by the presentation.  Arrow color represents the generator being followed.

Note that Cayley diagrams can be invariant to your particular choice of generator:

The shape of the Cayley diagram explains why $C_3$ is called a cyclic group, by the way!

With these tools in hand, let’s turn to more complex group structures.

Dihedral Groups

Cyclic groups have rotational symmetry. Dihedral groups have both rotational and reflectional symmetry. The dihedral group that describes the symmetries of a regular n-gon is written $D_{n}$. Let us consider the “triangle group” $D_{3}$, generated by a clockwise $120\circ$ rotation $r$ and a horizontal flip $f$.

With triangles, we know that three rotations returns to the identity $r^3 = e$. Similarly, two flips returns to the identity $f^2 = e$. Is there some combination of rotations and flips that are equivalent to one another? Yes. Consider the following equality:

Analogously, it is also true that $rf = fr^2$.

Definition 7. Some collection of elements is a generating set if combinations amongst only those elements recreates the entire group.

Cyclic groups distinguish themselves by having only one element in their generating set. Dihedral groups require two generators.

We can write each dihedral group element based on how it was constructed by the generators:

$D_n = \left\{ e, r, r^2, \ldots, r^n-1, f, rf, r^2f, \ldots, r^{n-1}f \right\}$

Alternatively, we can instead just write the presentation of the group:

$D_{3} = \langle r, f \mid r^3 = 1, f^2 = 1, r^2f = fr, rf = fr^2 \rangle$.

We can visualize this presentation directly, or as a more abstract Cayley graph:

The Cayley table for this dihedral group is:

This shows that $D_3$ is not abelian: its multiplication table is not symmetric about the diagonal.

By looking at the color groupings, one might suspect it is possible to summarize this $6 \times 6$ table with a $2 \times 2$ table. We will explore this intuition further, when we discuss quotients.

Until next time.

Wrapping Up

Takeaways:

• Finite groups can be analyzed with Cayley tables (aka multiplication tables).
• The same group can have more than one set-operation expressions (e.g., modular arithmetic vs. roots of unity vs. rotational symmetry).
• Generators, elements from which the rest of the set can be generated, are a useful way to think about groups.
• Group presentation is an alternate way to describing group structure. We can represent presentation visually with the help of a Cayley diagram.
• Cyclic groups (e.g., $C_3$) have one generator; whereas dihedral groups (e.g., $D_3$) have two.

Related Resources

• This post is based on Professor Macaulay’s Visual Group Theory lectures, which in turn is based on Nathan Carter’s eponymous textbook.
• Related to this style of teaching group theory are Dana Ernst’s lecture notes.
• If you want to see explore finite groups with software, Group Explorer is excellent.
• For a more traditional approach to the subject, these Harvard lectures are a good resource.

# Isotopy in Ambient Manifolds

Part Of: Analysis sequence
Followup To: An Introduction to Topology
Content Summary: 1300 words, 13 min read

Degenerate Geometries

Two lines can either be parallel, or not. There exist unending variations of both situation. But which is more common, on the average?

Consider two lines $y_{1} = m_{1}x + b_{1}$ and $y_{2} = m_{2}x + b_{2}$ When do we call these lines parallel? When their slopes are equal $m_{1} = m_{2}$. We can gain insight into the situation by mapping parameter space, where $m_{1}$ and $m_{2}$ form the horizontal and vertical axes respectively.

The red line does not represent one line, but an infinite set of parallel lines where $m_{1} = m_{2}$.

Suppose we start somewhere on the red line (with some pair of parallel lines)\$. Perturbation of the slopes of these lines corresponds to a random walk beginning at that point. The more you walk around on the plane, the more likely you are to stand in green territory (slopes whose lines are not parallel).

Definition 1. A situation holds generically if, by that perturbing its constituent properties, it tends to default to that situation.

This concept applies in many situations. For example:

• In two dimensions, two lines might be parallel, but generically intersect at a point.
• In three dimensions, two planes might be parallel, but generically intersect in a line.
• In linear algebra, a matrix might be singular is its determinant is equal to zero. But the determinant becomes non-zero on perturbation of individual matrix values. Thus, matrices are generically invertible.

This concept has also been called general position.

Manifold Intersection & Overflow

We now turn to the general science of intersection. The following observations hold generically:

• In two dimensions, a point and another point do not intersect.
• In two dimensions, a point and a line do not intersect.
• In two dimensions, a line and a line do intersect, at a point.
• In three dimensions, a line and another line do not intersect.
• In three dimensions, a line and a plane do intersect, at a point.
• In three dimensions, a plane and another plane do intersect, at a line.

Points, lines, planes… these are manifolds! Some of them infinitely large, but manifolds nonetheless! Let’s use the language of topology to look for patterns.

Each of the above examples contains two submanifolds $K$ and $L$ being placed in an ambient manifold $M$. We denote their intersection as $K \cap L$. Let us compare the dimensions of these three manifolds to the dimension of their overlap.

We represent $dim(M), dim(K), dim(L), dim(K \cap L)$ as $m, k, l, k \cap l$ respectively. Now our examples can be expressed as 4-tuples $(m, k, l, k \cap l)$:

• $(2, 0, 0, \varnothing)$
• $(2, 0, 1, \varnothing)$
• $(2, 1, 1, 0)$
• $(3, 1, 1, \varnothing)$
• $(3, 1, 2, 0)$
• $(3, 2, 2, 1)$

In four dimensions, would we expect two planes to intersect; and if so, what would we expect the dimension of the intersection? Put differently, what would we predict to be the value of $x$ in $(4, 2, 2, x)$?

If you guessed $x=0$, that the two planes intersect at a point, you have noticed the pattern!

Definition 2. Consider two submanifolds embedded in an ambient manifold $K, L \subseteq M$. The overflow is defined as $o = (k+l) - m$.

Theorem 3. Let $K, L \subseteq M$. The following properties are true, generically:

• If $o < 0$, the submanifolds do not intersect: $K \cap L = \varnothing$
• If $o \geq 0$, the intersection is non-empty $K \cap L \neq \varnothing$, and $dim( K \cap L )= o$

To see why this is the case, consider basis vectors in linear algebra. An m-dimensional space requires an m-dimensional basis vector. Submanifold dimensions are then “placed within” the ambient basis. If we try to minimize the overlap between two submanifolds, the equation for overflow falls out of the picture. 🙂

Overflow during Motion

We have considered overflow in static submanifolds. But what if we move one of them?

The following observations hold generically:

• In two dimensions, moving a point across a point do not intersect.
• In two dimensions, moving a point across a line intersect, at a point.
• In two dimensions, moving a line across another line intersect, across the entire line.
• In three dimensions, moving a line across another line intersect, at a point.
• In three dimensions, moving a line across a plane intersect, at a line.
• In three dimensions, moving a plane across another plane intersect, across the entire plane.

Compare these “motion” $(m, k, l, k \cap l)$ tuples against our previous tuples:

• $(2, 0, 0, \varnothing)$ vs. $(2, 0, 0, \varnothing)$
• $(2, 0, 1, 0)$ vs. $(2, 0, 1, \varnothing)$
• $(2, 1, 1, 1)$ vs. $(2, 1, 1, 0)$
• $(3, 1, 1, 0)$ vs. $(3, 1, 1, \varnothing)$
• $(3, 1, 2, 1)$ vs. $(3, 1, 2, 0)$
• $(3, 2, 2, 2)$ vs. $(3, 2, 2, 1)$

The dynamic overflow is one dimension larger than the static overflow. Why?

The way I like to think about it is, by moving K across time, you are effectively enlisting an extra dimension (time). A moving point starts to look like a string, etc.

Theorem 4.  Let $K, L \subseteq M$. Suppose we want to move K from one side to the other side of L. This crossing is said to be topologically possible if $K$ and $L$ do not intersect at any point during the transition. The possibility of a crossing depends on the overflow $o$:

• If $o < -1$, then generically, a crossing is possible (at all times, $K \cap L = \varnothing$)
• If $o \geq -1$, then generically, a crossing is not possible (at some time, $K \cap L \neq \varnothing$ and $dim( K \cap L )= o$

Towards Isotopy

Let’s put this theorem to work.

Example 5. In what $R^m$ is the following movement possible?

We know from physical experience that this is not possible in our three-dimensional universe $R^{3}$. But Theorem 4 says that, if the ambient dimension is four, then the overflow is $(1+1) - 4 = -2 < -1$, so crossing is possible.

Intuitively, this makes sense. In three dimensions, a point can “hop” over a line by moving into the “extra” dimension. Similarly, the line $K$ can cross over $L$ by moving into the fourth dimension.

We can generalize this notion of successful crossing as follows:

Definition 6. Imagine moving submanifold $K$ through an ambient manifold $M$. Let $K_{0}$ and $K_{1}$ represent the beginning and end positions as it travels throughout time $t \in [0,1]$. If $K_{t}$ never has self-intersection at any time $t$, we say $K_{0}$ is isotopic to $K_{1}$ in $M$, and write $K_{0} \sim_{M} K_{1}$

Isotopy is especially relevant to knot theory. A classic example is the trefoil knot, a simple kind of knot. The string trefoil knot composed of 1-dimensional string is not isotopic to the unknot (1-torus) in $\mathbb{R}^3$: this is why it is called non-trivial.

Example 7. Show how the trefoil knot can isotope to the unknot.

1. From $K_{0}$ to $K_{0.5}$: we can unwind the trefoil knot in $\mathbb{R}^4$: simply lift the top-left string up.
2. From $K^{0.5}$ to $K^{1}$: you only need $\mathbb{R}^3$ to finish the job. Simply pull the top loop down, and then untwist.

So knot theory is only interesting in $\mathbb{R}^3$! In $\mathbb{R}^2$, knots self-intersect. In $\mathbb{R}^4$, they are all isotopic to the unknot.

More Isotopy Examples

Let’s get some more practice under our belt.

Example 8. Consider a genus-two torus with its holes intertwined. What ambient manifold do we require to undo the knot?

We might imagine the solution isotope would require us to pull apart the “hands” directly. And it is true that we can successfully isotope in $M = \mathbb{R}^{6}$; after all $o = (2+2) - 6 = -2 < -1$.

But do we really need six dimensions to get this done? Or can we do better? It turns out that we only really need $M = \mathbb{R}^{3}$:

As the number of ambient dimensions increases, finding an isotope becomes increasingly easy. Thus, we often strive to find the smallest possible ambient manifold.

Example 9. Consider a genus-3 torus ($\Sigma^{3}$). If we isotope along its surface using $M = \Sigma^{3}$, is $K_{1} \sim K_{2}$? How about $L_{1} \sim L_{2}$

It is easy to see that $K_{1} \sim_{\Sigma^{3}} K{2}$. You just pull the circle left, along that surface of the torus.

But it takes some time to see that $L_{1} \nsim L{2}$. You might suspect you can just pull the blue circle over the middle hole. But that would require leaving the surface $\Sigma^{3}$. Thus $L_{1} \nsim_{\Sigma^{3}} L{2}$ (but $L_{1} \sim_{\mathbb{R}^{3}} L{2}$).

Related Materials

This post is based on Dr. Tadashi Tokeida’s excellent lecture series, Topology & Geometry. For more details, check it out!

# An Introduction to Set Theory

Part Of: Algebra sequence
Content Summary: 1800 words, 18 min read

Fundamentals of Sets

Definition 1. A set is a collection of distinct objects.

A few examples to kick us off:

• $MyFavoriteFruits = \left\{ apples, persimmon, pineapple \right\}$ represents the set of fruits which I prefer.
• $A = \left\{ 1, 2, 3, 4, 5 \right\}$ can represent, among other things, the fingers on my left hand.
• The set of natural numbers $\mathbb{N} = \left\{ 0, 1, 2, 3, 4, ... \right\}$.
• The set of integers $\mathbb{Z} = \left\{ ..., -2, -1, 0, 1, 2, ... \right\}$.

Sets are subject to the following properties:

• Order blindness: $\left\{2, 1, 3\right\}$ and $\left\{1, 2, 3\right\}$ expresses the very same set.
• Duplicate blindness. $\left\{ 1, 1, 2, 3, 3 \right\} = \left\{1, 2, 3 \right\}$. We will prefer to express sets with the latter, more compact, notation.
• Recursion. $\left\{ 1, \left\{2, 3\right\} \right\}$ is a perfectly valid two-element set, quite distinct from the three-element set $\left\{ 1, 2, 3 \right\}$.

Definition 2. Two sets A and B are said to be equal ($A = B$) if A and B contain exactly the same elements.

• Let $A = \left\{ 1, 2, 3 \right\}$ and $B = \left\{ 3, 1, 2 \right\}$. Then, $A = B$.
• Let $A = \left\{ 1, 2, 3 \right\}$ and $B = \left\{ 1, \left\{2, 3\right\} \right\}$. Then, $A \neq B$.

Definition 3. If an object x is an element of (i.e., member of) set S, we write $x \in S$. Otherwise, we write $x \notin S$.

• Let $PrimaryColors = \left\{ red, yellow, blue \right\}$. Then $yellow \in PrimaryColors$ means “yellow is an element of the set of primary colors”.
• $-1 \notin \mathbb{N}$ means “-1 is not an element of the natural numbers”.
• $1 \notin B = \left\{ 0, \left\{ 1 \right\}, \left\{ \left\{ 2 \right\} \right\} \right\}$. The element $1$ is not in $B$: only the set $\left\{ 1 \right\}$ is.

Definition 4. For some set $X$, its cardinality (i.e., size) $|X|$, is the number of elements in that set.

• Let $A = \left\{ 1, 2, 3, 4, 5 \right\}$. Then $|A| = 5$.
• Let $B = \left\{ 1, \left\{2, 3 \right\} \right\}$. Then $|B| = 2$. Note that cardinality only looks at “the outer layer”.

Definition 5. The empty set (i.e., the null set) $\varnothing$ is the set containing no elements.

• $\varnothing = \left\{ \right\}$.
• $| \varnothing | = 0$.
• $| \left\{ \varnothing \right\} | = 1$

Definition 6. Instead of listing all elements, set builder notation specifies rules or properties that govern the construction of a set. The notation takes the following form: ${ x | \normalfont{property of} x }$, where the | symbol is pronounced “such that”.

• $A = \left\{ 1, 2, 3, 4, 5 \right\} = \left\{ x \in \mathbb{Z} | x > 0, x < 6 \right\}$. In words “let A be the set of integers X such that x is greater than zero and less than six.”
• The set of rational numbers $\mathbb{Q} = \left\{ x / y \mid x \in \mathbb{Z}, y \in \mathbb{Z}, y \neq 0 \right\}$.

Sets defined by their properties are often called set comprehension. Such properties or rules are called the intension, the resultant set is called the extension.

• Let $A = \left\{ y | x \in \mathbb{R}, y = (x+1)(x-1) \right\}$ and let $B = \left\{ y | x \in \mathbb{R}, y = x^{2} -1 \right\}$. Here $A = B$, despite their use of different rules. We say that $A$ and $B$ have the same extension, but different intensions.

The intension-extension tradeoff denotes an inverse relationship between the number of intensional rules versus the size of the set those rules pick out. Let’s consider two examples to motivate this tradeoff.

Consider hierarchical addressing in computer architecture. Suppose we have $2^{6} = 64$ bits of computer memory, each bit of which is uniquely identified with a 6-bit address. Suppose further that our memory has been allocated to data structures of varying size. To promote addressing efficiency, a computer can adopt the following strategy: assign shorter addresses to larger variables.

We can also see the intension vs. extension tradeoff in the memory systems of the brain. Specifically, semantic memory is organized into a concept hierarchy. We might classify a Valentine’s day gift according to the following tree:

The number of objects classified as a RED_ROSE is clearly less than the number of objects classified as LIVING_THING. But as our extensional size decreases, the size of our intension (the number of properties needed to classify a RED_ROSE) increases.

Subsets and Power set

Definition 7. A set $A$ is a subset of another set $B$, written $A \subseteq B$, if every element of $A$ is also an element of $B$.

• Let $A = \left\{ 2, 3, 9 \right\}$ and $B = \left\{1,9,3,6, 2\right\}$. Then $A \subseteq B$ (recall that element order is irrelevant).
• Let $A = \left\{ 2, 3, 9 \right\}$ and $C = \left\{1,7, 3, 6, 2 \right\}$. Then $A \nsubseteq C$ (C does not contain 9).
• Is $\varnothing$ a subset of $\left\{2,3,9\right\}$?  Yes. For all $A$, $\varnothing \subseteq A$ is true.

Definition 8. A set $A$ is a proper subset of another set $B$, written $A \subset B$, if $A \subseteq B$ and $A \neq B$.

Definition 9. For a given set $A$, its power set $\mathbb{P} (A)$ is the set of all subsets of $A$.

• Let $A = \left\{ 0, 1 \right\}$. Then $\mathbb{P}(A) = \left\{ \varnothing , \left\{ 0 \right\} , \left\{ 1 \right\}, \left\{ 0, 1 \right\} \right\}$.

A power set can be constructed by the use of a binary tree, as follows:

As can be seen above, the total number of subsets must be a power of two. Specifically, if $|A| = n$, then $|\mathbb{P}(A)| = 2^n$.

It is important to get clear on the differences between the element-of ( $\in$ ) versus subset-of ( $\subseteq$ ) relations. Consider again $A = \left\{ 0, 1 \right\}$ and its power set $\mathbb{P}(A) = \left\{ \varnothing , \left\{ 0 \right\} , \left\{ 1 \right\}, \left\{ 0, 1 \right\} \right\}$

• $A \in \mathbb{P}(A)$. But $\left\{A\right\} \not\in \mathbb{P}(A)$. The $\in$ relation requires the brackets match.
• $\left\{A\right\} \subseteq \mathbb{P}(A)$. But $A \nsubseteq \mathbb{P}(A)$. The $\subseteq$ relation requires the “extra bracket”.

Probability theory is intimately connected with the notion of power set. Why? Because many discrete probability distributions have $\sigma$-algebras draw from the power set of natural numbers $\mathbb{P} ( \mathbb{N} )$.

Cartesian Product, Tuples

Definition 10. Given two sets $A$ and $B$, their Cartesian product $A \times B$ is the set of all ordered pairs $\langle a, b \rangle$ such that $a \in A$ and $b \in B$. Note that, unlike the elements in a set, the elements of an ordered pair cannot be reordered.

• Let $A = \left\{1, 2, 3 \right\}$ and $B = \left\{4, 5, 6\right\}$. Then $A \times B = \left\{ (1, 4) , (1, 5), (1,6), (2, 4), (2, 5), (2, 6), (3, 4), (3, 5), (3, 6) \right\}$.

We can represent this same example visually, as follows:

• Contrast this with $B \times A = \left\{ (3, 1) , (4, 1), (2, 4), (1, 3) \right\}$. Thus, $A \times B \neq B \times A$. This is because elements within ordered pairs cannot be rearranged.
• Note that $|A \times B| = 9 = |A_{1}| \times |A_{2}|$. In combinatorics, this observation generalizes to the multiplication principle.
• The real plane $\mathbb{R}^2 = \mathbb{R} \times \mathbb{R}$ is a well-known example of a Cartesian product.

Definition 11. Given n sets, $A_{1}, A_{2}, \ldots , A_{n}$, their Cartesian product is the set of all n-tuples.

• Let $A = \left\{1, 2\right\}$, $B = \left\{a, b\right\}$ and $C = \left\{ 100 \right\}$. Now $A \times B \times C = \left\{ (1, a, 100) , (1, b, 100), (2, a, 100), (2, b, 100) \right\}$.

Linear algebra is intimately connected with the Cartesian product operation. Why? Because n-tuples are strongly related to n-dimensional vectors.

Intersection and Union

Definition 12. The intersection of two set $A$ and $B$, written $A \cap B$, is the set of elements common to both sets.

• Let $A = \left\{ 2, 4, 9 \right\}$ and $B = \left\{1, 2, 3, 6, 9 \right\}$. Then $A \cap B = \left\{ 2, 9 \right\}$.
• Let $A = \left\{ 2, 4, 9 \right\}$ and $B = \left\{1, 3, 6, 8, 10\right\}$. Then $A \cap B = \varnothing$.
• Let $A = \left\{ 2, 4, 9 \right\}$. Then $\varnothing \cap A = A \cap \varnothing = \varnothing$.

Definition 13. The union of two sets $A$ and $B$, written $A \cup B$ is the set of all elements that are in $A$ or $B$, or both.

• Let $A = \left\{ 2, 4, 9 \right\}$ and $B = \left\{1, 2, 3, 6, 9 \right\}$. Then $A \cup B = \left\{ 1, 2, 3, 4, 6, 9 \right\}$.
• Let $A = \left\{ 2, 4, 9 \right\}$. Then $\varnothing \cup A = A \cup \varnothing = A$.

Venn diagrams represent sets as enclosed areas in a 2D plane. If two (or more!) sets have shared elements, their areas overlap. We can use this technique to visualize sets and their overlap:

We can also use Venn diagrams to represent our intersection and union relations:

Note that $|A \cup B| = |A| + |B| - |A \cap B|$. This makes sense in light of the Venn diagram. Adding the cardinality of both sets counts the elements that exist in the middle section twice. To avoid this, we subtract the cardinality of the intersection. In combinatorics, this formula is generalized by the inclusion-exclusion principle

Difference and Complement

Definition 14. Given two sets $A$ and $B$, their difference $A \setminus B$ is the set of elements in $A$ but not also in $B$.

• Let $A = \left\{ 1, 2, 3, 4, 5, 6 \right\}$ and $B = \left\{0, 2, 4, 6, 8 \right\}$. Then $A \setminus B = \left\{ 1, 3, 5 \right\}$ and $B \setminus A = \left\{ 0, 8 \right\}$.
• Let $A = \left\{ 1, 2, 3, 4, 5, 6 \right\}$ and $B = \left\{7, 8, 9, 10 \right\}$. Then $A \setminus B = A$ and $B \setminus A = B$.

Definition 15. Given two sets $A$ and $B$, the symmetric difference $A \triangle B$ is the set of elements in $A$ or $B$, but not both.

• Let $A = \left\{ 1, 2, 3, 4, 5, 6 \right\}$ and $B = \left\{0, 2, 4, 6, 8 \right\}$. Then $A \triangle B = \left\{ 0, 1, 3, 5, 8 \right\}$.

Definition 16. In many set problems, all sets are defined as subsets of some reference set. This reference set is called the universe $U$.

• Let $A = \left\{ 1+i, 12-8j, 3+0i \right\}$ and let its universe be the set of complex numbers $\mathbb{C}$. It is true that $A \subseteq \mathbb{C}$.

Definition 17. Relative to a universe $U$, the complement of $A$, written $\overline{A}$ is the set of all elements of $U$ not contained in $A$.

• Let U be the set of positive integers less than 10: $U = \left\{ x | x \in \mathbb{Z}^{+}, x < 10 \right\}$ and $A = \left\{ 1, 2, 3, 4, 5 \right\}$. Then $\overline{A} = \left\{ 6, 7, 8, 9 \right\}$.

We can again represent these relations graphically, via Venn diagrams:

Takeaways

Let me summarize this post in terms of our 17 definitions 🙂

• Def 1-5 introduced the notion of set, set equality, the element-of operator, cardinality (set size), and empty set.
• Def 6 introduced set builder notation, and the intension-extension tradeoff.
• Def 7-9 introduced subset, proper subset, and power set.
• Def 10-11 introduced Cartesian product, ordered pairs, and n-tuples.
• Def 12-13 introduced intersection (“and”) and union (“or”), as well as Venn diagrams.
• Def 14-17 introduced difference, symmetric difference (“xor”), and complement (“not”).