Math Practice: Basic Set Theory

 
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.

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set theory, deck of cards - magoosh
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: AB. When you combine two or more sets, the new set includes the elements from each set. So here we would have:

AB = [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 AB. So using the same sets as above:

AB = {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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