# Relations

An **ordered pair** is a pair of things, but in this case we actually care which is first and which is second. You know what they say about second: first loser.

To write an ordered pair, we write down the two things separated by a comma, then put parentheses around the lot. Not curly brackets, mind you. We're back to parentheses now. This may seem confusing, but at least we haven't thrown straight brackets into the mix as well. Yet.

### Sample Problems

(2, 3) is an ordered pair. The first thing is 2 and the second thing is 3.

(*a*, *b*) is also an ordered pair. The first thing is *a* and the second thing is *b*. Hey, just like in the alphabet!

A **relation** is a set of ordered pairs. A relation can be finite or infinite. However, infinite relations are in complete denial. If you were to ask one of them, "How are you doing?" they would probably answer, "I'm finite."

### Even More Sample Problems

{(1, 10), (9, 11)} is a relation. This is a set containing two ordered pairs.

{(1, 2), (2, 4), (3, 6). . .} is an infinite relation. The relation contains infinitely many pairs, thanks to those repeating dots.

{(1, 1), (1, 2), (1, 3), . . .} is an infinite relation.

{(2, 3), (3, 4), 5, (2, 1)} is *not* a relation, however. This set contains the element 5, which is not an ordered pair. If you wanted to sneak into this group, 5, couldn't you have at least worn some parentheses to try to blend in? Where's the effort?

The empty set ∅ may also be considered a relation: the empty relation. We're not sure, but we have a feeling this would have been Jean-Paul Sartre's favorite relation.

The **domain** of a relation is the set whose elements are the first things from the relation's ordered pairs. We admit that's a little wordy, so we'll do what we do best: say it with numbers. Here's an example, coming at ya...

### Sample Problems

The domain of the relation {1, 10), (9, 11)}

is the set {1, 9}.

Got it? We pulled the first element from the first set and the first element from the second set. If it helps, think of the "main" part of "domain": we pull the "main," or "first," parts out of our ordered pairs. It would be helpful if the word was "dofirst," but unfortunately we don't live in a perfect world.

The **range** of a relation is the set consisting of the second things from the relation's ordered pairs. Makes sense.

### Sample Problems

The range of the relation {(1, 10), (9, 11)}

is {10, 11}.

If the "main" trick isn't doing it for you, try thinking that "d'' for domain comes before "r'' for range in the alphabet. The domain is the set of first things from the ordered pairs, and the range is the set of second things. Hopefully the alphabet doesn't still give you fits.

Sometimes we can use an equation to express the connection between the first and second things in the ordered pairs of a relation. This is good news, as you have probably been going through equation withdrawal during the early part of this unit.

In the relation

{(1, 2), (2, 3), (3, 4), . . . }

we can see that, in each ordered pair, the second number is one greater than the first number. Pattern alert, people. We could express this connection by saying

(second number) = (first number) + 1.

However, you know how much we hate to use big long words or phrases in equations. It cramps our style. In symbols, we could express the connection by saying that *x* is the first thing in the ordered pair, *y* is the second thing in the ordered pair, and

*y* = *x* + 1.

Since *x* comes before *y* in the alphabet, *x* corresponds to the first thing in the ordered pair, and *y* corresponds to the second thing in the ordered pair.

first thing | second thing |

in Domain | in Range |

x | y |

This is easy to remember, because there is an *x* in "domain" and a *y* in "range." Shhh. Humor us just this once.

### Sample Problems

Write an equation that expresses the connection between *x* and *y* in the following relation:

{(1, 1) , (2, 4), (3, 9), (4, 16)}.

As we already established, *x* refers to the first number in each ordered pair and *y* refers to the second number in each ordered pair. Since the second number in each pair is the square of the first, the equation we're looking for is

*y* = *x*^{2}.

You didn't think squares and square roots were going away any time soon, did you? They're here to stay, we're afraid. Like that guy your mom has been seeing who thinks everything he says is hilarious. We are referring, of course, to your dad.

### Sample Problems

Write an equation that expresses the connection between *x* and *y* in the following relation:

{(-1, 1), (-2, 4), (-3, 9), (-4, 16)}.

The equation we're looking for is the same as in the last example:

*y* = *x*^{2}

As we can see from this example, it is possible to have two relations where the connection between *x* and *y* is described by the same equation, but where the two relations aren't the same. We can imagine how that might get confusing or misleading, but mathematicians wouldn't have decided to express things this way if there wasn't a good reason for it. They're very calculating. Literally.

It's also possible to go from an equation to a relation. However, when we do this we have to specify which pairs in the relation we want. If we don't specify, then we get all the pairs that could possibly work. We are all for getting free swag, but only the pairs we need here will be just fine.