Intro to Set Theory

In algebra and calculus, you’ll sometimes come across sets of numbers, which are usually denoted with special symbols like curly brackets and the “element” symbol.

 
Even though at first glance these unfamiliar symbols can be intimidating, you’ll soon see that set theory is a helpful tool that is not too difficult to master.

When we talk about sets in mathematics, we are just talking about collections of objects. For example, I could have a set called “A” which contains the even numbers from 2 to 8.

Or I could have a set “B” containing all integer multiples of 5.

We can create sets for any collection of objects or numbers. So if you were interested in writing down the ages of everyone in your class, you could express those numbers together as a set. In both of the example sets A and B, notice that we started by writing the name of the set, followed by an equals sign, and then wrote the elements of the set, separated by commas, between a pair of curly brackets. The numbers in a set are referred to as the elements of that set, and we use the element sign \((\in )\) to show that a number belongs to a set. For example, we can say that “2 is an element of A,” because 2 is one of the numbers in set A. Likewise, we can say that “5 is an element of B.”

Sometimes it is helpful to note that a particular number does not belong to a set. To denote this, we just write the element symbol with a strikethrough. For example, the number 2 is not in set B, so we would write \(2\not \in B\). Likewise, the number 5 is not in set A, so we can write \(5\not \in A\).

One of the interesting properties of sets in mathematics is that all sets include something called “the empty set.” The empty set is exactly what it sounds like; it has no elements and iis denoted using this symbol: \(∅\). You know how whenever you have \(x\) in an equation, it’s understood to mean you have one \(x\)? The 1 is understood, and therefore doesn’t need to be written. In the same way, it is understood that all sets contain the empty set. One way to think about this is to say “the set A has 2, 4, 6, 8, and nothing else.” The “nothing else” is the empty set, so \(∅\in A\). And of course, \(∅\in B\) as well.

We’ve already talked about using the element symbol to show that one number is contained in a set. If we wanted to show that multiple items belonged to a set, we could use the subset symbol, which looks like this: \(\subseteq \). For example, we can say that the numbers 2, 4, and 6 are all in A by writing \({2, 4, 6} \subseteq A\). Additionally, let’s say we had a set C which contained only positive multiples of 5.

We could write that C is a subset of B because everything in C is included in B.

Because we said the empty set is understood to be in every set, we can write that the empty set is a subset of every set too.

Let’s say we have a set called \(A*\) which is identical to A.

Because everything in \(A*\) is included in A, we can say that \(A*\subseteq A\). Likewise, everything in A is included in \(A*\), so \(A\subseteq A*\).

When a subset is not equal to the entire set, we say that it is a proper subset and we can use this symbol: \(\subset \). This is kind of like using a less-than or greater-than symbol instead of a less-than-or-equal-to, or greater-than-or-equal-to symbol. A subset symbol with an underline is saying “subset or equal to.”

Earlier we wrote \(C\subseteq B\), which is true. But we can also write \(C\subset B\), since the set C does not contain all of the elements of B.

A similar term you may come across is uperset, which is basically the opposite of a subset. We said earlier that the set \({2, 4, 6}\) is a subset of A. Another way of saying this is that A is a superset of \({2,4,6}\). We write this with a mirrored subset symbol: \(A \supseteq {2,4,6}\).

Let’s go through a couple quick examples of what we’ve learned so far. For the set \(D=\){\(1, 2, 3, 4, 5\)}, which of the following statements is true?

The true statement is the one on the left: 4 is an element of D, because 4 is included in the set D.

Let’s say we have a set \(E =\){\(1, 2, 3, 4, 5, 6\)}. Which of the following statements is true about E?

Even though D and E are similar, E includes the number 6, unlike D. Because not all elements of E are in D,the last option is correct. E is not a subset of D.

Sometimes it can be helpful to visualize sets and their overlaps using Venn diagrams. Most commonly, Venn diagrams look like two overlapping circles, but they can feature three or more circles if necessary, and some circles may lie fully within others!

Let’s consider a regular two-circle Venn diagram, where the left circle is a set called L and the right circle is a set called R.

If we wanted to label the small area of overlap between L and R, we would call it “the intersection of L and R,” which is written with an upside-down U-shape: L\(\cap\) R. Intersections represent only the elements which are common to both sets. This is kind of like how we call the area where two roads cross each other an intersection; that area belongs to both roads!

If, on the other hand, we wanted to talk about everything in L together with everything in R, we would say “the union of L and R,” which is written with a U-like symbol: L \(\cup\) R. Unions are used to unite the elements from two sets.

Like some operations in mathematics, union and intersection follow the commutative, associative, and distributive properties. Let’s quickly review what each of those mean. The commutative property tells us that \(A\cap B=B\cap A\), and similarly that \(A\cup B=B\cup A\).

The associative property tells us that \((A\cap B)\cap C=A\cap (B\cap C)\); likewise, \((A\cup B)\cup C=A\cup (B\cup C)\).

Finally, the distributive property lets us rewrite \(A\cup (B\cap C)\) as \((A\cup B)\cap (A\cup C)\). Similarly, \(A\cap (B\cup C)=(A\cap B)(A\cap C)\).

Feel free to pause the video now and take a moment to write each of these properties down.

Another helpful term to know with set theory is called the universal set, which is often called U. It contains all elements of sets in consideration without repeats, and usually it comes into play when Venn diagrams are used. For example, this Venn diagram has sets G and H, as well as some elements which are neither in G nor H. Here, U contains all of the elements featured in the diagram: \(U=\){\(-3,-2,-1,0,2,4,5,6,7\)}.

Sometimes it can be helpful to refer to all elements except those in a particular set. This can be done using the complement of a set. There are different ways to denote the complement of a set; some people use an apostrophe, while others may use a horizontal bar over the set, and still others use the letter C. In this diagram, we could show the complement of G as the set of all elements not in G.

Similarly, the complement of H can be found by selecting all elements in the universal set which are not in H.

Similar to the idea of complements, sometimes you need to find elements which belong to one set but not to another. To do this, we use a technique called “set subtraction,” which is denoted using a backslash. For example, G minus H can be written like this (G∖H) and includes the elements that are in G but not in H. This would ignore the elements \(0\) and \(–\)2, leaving us with only \(-1\) and \(-3\).

Similarly, H minus G would include all elements of H but those that are also in G.

Let’s wrap up by naming some common sets that you’ll likely come across in your assignments. We know that the natural numbers, which are also called the counting numbers, are whole numbers starting with 1. This set of numbers is abbreviated with a fancy letter N.

The integers are whole numbers, including both positives and negatives, as well as zero. The set of integers is written as a fancy Z, which comes from the German word “Zahlen” for numbers.

The set of rational numbers, which are all numbers that can be expressed as fractions of integers and are not endless in their digits like pi, is called Q. The rational numbers are denoted by a Q because they can all be written as quotients.

The set of real numbers includes all rational and irrational numbers, and is denoted with the letter R.

Finally, the set of complex numbers includes all real and imaginary numbers, and is denoted with a C.

Note that because all natural numbers are integers, all integers are rational numbers, all rational numbers are real, and the real numbers are a subset of complex numbers, we can write it this way:

Hearing all of this information now probably seems like a lot, so feel free to watch through this video a second time if necessary. As with all things though, this will come to you more naturally with practice. Now that you’ve got a foundation for understanding set theory, you’re ready to try some example problems on your own!

I hope this video was helpful. Thanks for watching, and happy studying!