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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.

This lesson will define relations, partial order relations, and total order relations. We will look at examples of these types of relations in both a real world context and a mathematical context to facilitate understanding of these concepts.

Order Relations

Suppose there's a video game coming out today. At the game store, you run into a huge line of people. You realize you're going to be there for awhile, so let's give you something to think about while you're waiting in line; how about the line itself? It just so happens that you can describe the relationship between people's spots in line with an interesting mathematical concept. What? No way! You have to admit that you're intrigued.

In mathematics, if S is a set of elements, then a relation on S, call it R, is a set of ordered pairs, (x, y), where both x and y are elements of S, and x is related to y by some rule.

If you let S be the set of all the people in line, and represent each person by their place in line, then you can define a relation, R, on S where (x, y) is in R if x and y are people in line, and x is either in front of y or is y.

For instance, consider person 3 and person 10 in line. The ordered pair (3,10) would be in the relation R, because both people are in line, and person 3 is in front of person 10.

Well that's pretty neat! However, the line still hasn't moved, so let's explore this concept a bit further to keep you occupied.

Partial Order Relations

A relation T on a set M is called a partial order relation when it satisfies the following properties:

It's reflexive: (x, x) is in T for every x in M.

It's antisymmetric: If (x, y) is in T and (y, x) is in T, then x = y.

It's transitive: If (x, y) is in T and (y, z) is in T, then (x, z) is in T.

Oh boy, that's a lot of abstractness. Thankfully, the line relation, R, is a partial order relation, so you can use it to make these properties more understandable.

Reflexive property: In the line relation, (x, y) is in R if x is in front of y or x is y. Because of the latter part of that statement (x is y), you have that (n, n) is in the relation R, for all person n in line, because person n is person n. Therefore, (n, n) is in R for all n in S. For example (7, 7) is in R because person 7 is person 7.

Antisymmetric property: Suppose (x, y) and (y, x) are in R, so person x is in front of person y or is person y, and person y is in front of person x or is person x. Since they can't both be in front of the other, it must be the case that person x is person y. That is, x = y. You get that if (x, y) and (y, x) are in R, then x = y. x = y because they can't both be in front of each other.

Transitive property: If person x is in front of person y in line, then (x, y) is in R. Now, if person x is in front of person y, and person y is in front of person z, then person x is certainly in front of person z. Thus, you have that if (x, y) and (y, z) are both in R, then (x, z) is also in R. For example, if (3, 4) and (4, 5) are in R then (3, 5) is in R because person 3 is in front of person 5.

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Sigh, the line still isn't moving. At least that means you have time to consider total order relations. A relation T on a set M is a total order relation if it is a partial order relation (reflexive, antisymmetric, and transitive), and it satisfies one more property: for any two elements, x and y, in M, (x, y) is in T or (y, x) is in T. In other words, every element in M is related to every other element in M in some way.

Hmm. . . what do you think? Is our line relation, R, a total order relation? Well, if you grabbed two people from line, say person x and person y, one of them has to be in front of the other, so it must be the case that (x, y) is in R or (y, x) is in R. Ah-ha! It is a total order relation!

Mathematically Speaking

You know that our line relation is a partial order relation and a total order relation. Let's try to put the line relation in mathematical terms. You are representing each person in line with their place in line, so the set of people in line is S = {1,2,3,4,5, and so on}, so S is the set of positive integers. Also, the relation, R, consists of the ordered pairs (x, y), where x is in front of y or is y. In other words, x â‰¤ y.

Basically, our line relation R is the mathematical relation R on the set, S, of positive integers, consisting of the ordered pairs (x, y) where x â‰¤ y. Furthermore, this mathematical relation is both a partial order relation and a total order relation. It's so neat how we can look at this in either a real world context or in a mathematical context!

Lesson Summary

If S is a set of elements, then a relation on S, call it R, is a set of ordered pairs, (x, y), where both x and y are elements of S, and x is related to y by some rule. A relation, R, on a set S is a partial order relation if it is reflexive, antisymmetric, and transitive. R is a total order relation if it is a partial order relation, and it satisfies the property that for any two elements, x and y, in S, (x, y) or (y, x) is in R.

As you've seen with our line relation, these types of relations show up in the real world often, and you can think of them in either real world terms or mathematical terms to better analyze the scenario. Speaking of which, the line just started moving! Hopefully, this lesson made your wait a little more interesting.

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