# Binary Relations: Definition & Examples

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• 0:03 What Are Binary Relations?
• 2:07 Another Example of…
• 3:03 An Example in Mathematics
• 5:02 Lesson Summary
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Lesson Transcript
Instructor: Laura Pennington

Laura received her Master's degree in Pure Mathematics from Michigan State University. She has 15 years of experience teaching collegiate mathematics at various institutions.

In this lesson, we'll define binary relations. We'll look at examples in both a real-world context and a mathematical context to illustrate the concept of binary relations and to practice working with these types of relations.

## What Are Binary Relations?

Do you have a book that you keep your phone numbers in - a list of names of people you know along with their phone numbers? If so, then here's an interesting fact: that list is a binary relation! Wait, a what? That's a pretty fancy name. What the heck does it mean?

Technically speaking, in mathematics a binary relation, from a set M to a set N, is a set of ordered pairs, (m, n), or m and n, where m is from the set M, n is from the set N, and m is related to n by some rule. Confused yet? Let's see if we can put this into terms that we can better understand using your list of names and phone numbers.

If we let M be the set of all of the names of the people on your phone list, and we let N be the set of all of those phone numbers on the list, then your list relates each name in the set M to a number in the set N. Therefore, your list, which we'll call L, is a binary relation from the set M to the set N.

To clarify further, say that your friend Andy Smith has phone number 123-456-7891. Then the ordered pair (Andy, 123-456-7891) would be in the relation L, because Andy is in set M (the names), 123-456-7891 is in the set N (the phone numbers), and Andy is related to 123-456-7891 by the rule that 123-456-7891 is Andy's phone number.

Hmm. . . it's starting to make sense. Basically, binary relation is just a fancy name for a relationship between elements of two sets, and when an element from one of the sets is related to an element in the other set, we represent it using an ordered pair with those elements as its coordinates. Bingo! That's a binary relation!

That seems simple enough. Of course, these relations can be simple, as in our phone number example, or they can be more complicated. It all depends on the sets involved and the rule relating those sets. Let's explore this concept a bit further.

## Another Example of Binary Relations

In our phone number example, we defined a binary relation, L, from a set M to a set N. We can also define binary relations from a set on itself. That is, we call a relation, R, from set M to set M, a binary relation on M.

For example, suppose you are at a work event with your coworkers, and a team building activity requires everyone at the event to pair up with someone that has the same hair color as them. If we let Q be the set of all of the people at the event, then this pairing off is a binary relation, call it R, on Q. Basically, R is the binary relation that consists of the ordered pairs (q1, q2), where q1 and q2 are elements of Q, and q1 has the same hair color as q2.

This is becoming more and more clear. Now that we are more familiar with the concept of binary relations, let's take a look at a binary relation in mathematics.

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