I like to cheat at cards, mainly because I am really bad at reading peoples’ faces during poker. To do this, I think about the cards as a whole and what cards are still available in the set. Ah, the word *set* brings up a good point. In math, the **set theory** involves a collection of values or even objects called *elements*. This theory of sets leads to some pretty interesting math, and allows us to define a lot of things.

_{Image by Jalil Shams}

## What Is a Set

The elements in a set can be anything from numbers, to symbols, or even objects. For example, a set of cards can be represented by *A* = {♠, ♣, ♥, ♦}. Notice the elements are distinct and separated by commas. Furthermore, the set itself is set off by brackets. A set could (and most often does) contain numerical values, like *A* = {2, 4, 6, 8}.

## How Do Sets Work?

The nice thing about sets is that they can be manipulated, combined, removed, and more. There are two primary manipulations that are worth committing to memory: *union* and *intersection*. As examples, lets consider the following sets:

*A* = {1, 2, 3, 4}

*B* = {2, 4, 6, 8}

### Union

A union of sets means combining the elements of two or more sets into a new, more inclusive set. A union of sets is notated like this: *A* ∪ *B*. When you combine two or more sets, the new set includes the elements from each set. So here we would have:

*A* ∪ *B* = [1, 2, 3, 4, 6, 8}

The set is now bigger because it contains elements from both sets. Notice that the union combines the sets but does not duplicate values. For example, both *A* and *B* contain the values 2 and 4, but we do not need to duplicate it within the union. So, instead of having two 2’s and two 4’s, we only have one of each.

### Intersect

When sets intersect, we are only concerned with the elements that occur in *all* sets. In this sense, we are combining the sets in an exclusive sense—all elements except the common ones are eliminated. An intersection is notated as *A* ∩ *B*. So using the same sets as above:

*A* ∩ *B* = {2, 4}

In this case the intersection is smaller, but that is not always the case given that sets can be different sizes. Again, the intersection does not contain duplicate values.

## Takeaways for Set Theory

- A set is a collection of objects, things, or values called elements.
- Sets can be mathematically manipulated like many other objects.
- The two primary functions involving sets are union and intersection.
- A union of sets combines the elements that are common to the two or more sets while intersection includes only the elements that are common to both sets.
- Both union and intersection have properties that can be exploited to better understand the elements and how they relate to one another, especially when it comes to probability.

I hope that this helped clarify the basics of set theory, but check this out if you want to know more. Have some happy math!

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