Image by Jorgelrma

Statistics starts with probability, such as the probability that I will win the lottery (surprise, it’s not high). But when you open your textbook, you see a lot of weird symbols and theorems. Those weird symbols are part of **set theory**. There are some essential parts of set theory that you should sear into your mind if you want to interpret the language of probability.

Before we get started, let’s start with what set theory uses. A *set* is a collection things, called *elements*. Now, these elements can be anything like numbers, colors, characteristics, or even ♥ and ♣. For the sake of all that is mathematical and logical, we will use numbers and the occasional variable. Sets are usually set off by brackets like A_{i} = {1, 2,i 3,…, *i*}. Now, let’s talk about the top ten topics about that you should memorize like the Dickens.

### 1. Standard Sets

There are a few sets that occur naturally in math. Each of these sets has predefined components, much like a deck of cards has predefined suits and each suit contains the same cards.

There is a set of natural numbers designated by the symbol ℕ. Remember that a natural number is any positive, whole number. This set is ℕ = {1, 2, 3, …}

The next standard set is for integers. Integers are any whole number, whether it is positive or negative. So this set is ℤ = {…, -2, -1, 0, 1, 2,…}

Then there is the standard set of all rational numbers. A rational number is a number that can be in the form p/q where p and q are integers and q is not equal to zero. So ℚ = {9/4, -1, 0.5, 4/6, …}

Next is the set of all real numbers. A real number is any rational or irrational number that can be placed on a number line like 9/4 or even π. The symbol for the real numbers set is ℝ. So we would write the set like ℝ = {3.14…, 2/5, 8, …}

One that you really need to know is the set of complex numbers. These are numbers that include an imaginary component. This means that they *cannot* be found on a number line because they include a √-1 component. They are written in either radical or complex form. So the set is something like ℂ = {√-4, 3 + 4i, …}.

The last one is a set that crops up a lot, the universal set. Its symbol is *S* and it is the set that contains every value of the population that you are interested in and the subset.

### 2. Symbols

So far, you may have noticed that set theory uses some funky symbols like ℚ. Well, there are some more that pop up a ton, so let’s take a walk through some of the basic symbols.

∈ is the symbol includes or “is in”. For example A ∈ (-3, 3), so the set is A = {-3, -2, -1, 0, 1, 2, 3}.

∣ means “such that”. This one is a little odd because it contains instructions for the set. E.g., A = {x^{2} ∣ x = ℕ}. The set would be A = {1, 4, 9, 16, …}

Then there are *subsets *and *superset*. A subset is a set in which all the values come *from* another set. When we write A ⊂ B, all the elements in A are contained in B. That makes B the superset, or the one that contains all the elements of A and then some. This looks like B ⊃ A.

### 3. Union

Sometimes sets overlap. For example, say that you have sets A = {1, 2, 4, 9} and B = {2, 4, 6, 8}. So we can say that there is a subset where they overlap. That overlap is called **union**. To write this in set notation, we say A ∪ B = {1, 2, 4, 6, 8, 9}. Notice that I didn’t write two 2s or two 4s. That is because we are only concerned about which values overlap, not how many there are of each.

In a diagram, it would look like

### 4. Intersect

Sometimes, we are only concerned with the overlap of two or more sets. If A = {1, 2, 3, 4, 5, 6} and B ={2, 4, 6} the overlap would be 2, 4, and 6. We call this an **intersect** in set theory and we write it like A ∩ B. The diagram below is an example

The concept of the intersection of two (or more) sets is a useful idea for conditional probability.

### 5. Complement

Sometimes, we are concerned with what is *not* in the set, or what the set takes away from the universal set. We can consider the **complement** of a set to be the elements that are in the universal set, *S* but not in the set itself.

For example, let’s say you have a six-sided die. The universal set is *S* = {1, 2, 3, 4, 5, 6}. Let’s take it a step further and say that set A within the universal set is A = {2, 4, 6}. That means that 1, 3, and 5 are left over in the universal set. These elements make up the complement of A (A^{c}) because they are in the universal set, but *not* in A, or A^{c} = {1, 3, 5}

A diagram would look something like

### 6. Difference

You are familiar with regular subtraction, like 5 – 2 = 3. But sets can be subtracted from one another too. However, subtraction is a little different. In the case of A – B, subtraction means not only taking out B, but also the components that intersect A and B.

So, if we had A = {1, 2, 3, 4, 5, 6} and B = {2, 4, 6, 8} then A – B = {1, 3, 5}. You don’t have to worry about the extra elements in B since they were not in A in the first place. A diagram would look like

On a side note, the subtraction of sets is not entirely like the subtraction of numbers. A – B does not equal B – A because the sets contain different elements, so you may not be removing the same elements from both sets.

### 7. Disjoint

So far, we have been discussing sets that overlap or take away from overlapping areas. But what if the sets do not overlap or have no elements in common? These are called **disjoint** sets. This means that if were were to think about an overlap (A ∪ B) there would be none or A ∪ B = Ø.

This is a little easier to see with a diagram like the one here

It may not make sense to talk about sets that *don’t* overlap, but it is useful when we consider mutually exclusive events.

### 8. Partition

Kentucky (the state where I live) is a decent size for a state. But Kentucky is divided into 120 counties. You could go so far as to say that the state is partitioned into counties. When a set is divided into distinct and disjoint sets, each of those sets are **partitions** of the larger (usually universal) set.

In a partitioned set, the union of the partitions gives the overall set. So in the diagram below, A_{1}, A_{2}, A_{3}, and A_{4} are partitions of the overall set A.

### 9. De Morgan’s Law

Just as you can subtract sets from one another and you can transform them in other ways too. **De Morgan’s Laws** are the theoretical way that sets can be transformed. the complement of the union of two sets is the same as the intersection of their complements; and the complement of the intersection of two sets is the same as the union of their complements.

Good gravy that is technical.

Let’s think through it a little. Think of A_{1} ∩ A_{2} is the intersection of two sets. So, what is the complement of the intersection? Everything else! So, the complement of an intersection is the union of remaining complements. It is typically written as

(A_{1} ∩ A_{2})^{c} = A_{1}^{c} ∪ A_{2}^{c}

The opposite is also true

(A_{1} ∪ A_{2})^{c} = A_{1}^{c} ∩ A_{2}^{c}

### 10. Distributive Law

There are other ways to interpret sets. One basic way that is used for probability proofs and calculations is the **distributive law**. Using the distributive law, you can relate sets through union and intersection. The law states that taking the union of a set to the intersection of two other sets is the same as taking the union of the original set and both the other two sets separately and then taking the intersection of the results.

It is written as

A ∩ (B ∪ C) = (A ∪ B) ∩ (A ∪ C) or A ∪ (B ∩ C) = (A ∩ B) ∪ (A ∩ C)

Check out this post to visualize these sets better.

## The Takeaways

Set theory looks like a complex series of ideas and symbols, but it doesn’t have to be. We have discussed a few of the most important topics, especially union and intersect. I do want to tell you that the using Venn diagrams will help you to understand most of the topics and theorems in probability and set theory, just as we have used them here.

If you can get a hold of these topics, then you should be pretty *set* (pun intended) for anything in set theory that gets thrown at you. Happy statistics!

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