College Algebra: Intro to Relations and Functions

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You know, when you think about connections between one thing and something else, something's proportional with something else, or you're given a connection with an equation--for example, a line: y = 3x + 2--if you give me x, then I can figure out what y is. Well in fact that is a particular example of a much more general circumstance, something known as a relation. And a relation is nothing more than basically connections between two classes or two collections of things. And so, really, when you think of a relation, which is a very abstract idea, we're really thinking of a set of ordered pairs. So, I'm going to define to you what a relation is, first of all, in very abstract terms, and then take a look at some real examples of this.
A relation is really just a connection and therefore a set of ordered pairs. Here's an example: Set ({), and now I'm going to give you ordered pairs--{(3, -1), (2, 4), (3, 0)}. So, this whole thing is considered a relation, because, basically, for each first person here, I associate a second person. So the 3 here is associated with a -1; the 2 here is associated with 4; the 3 here is associated with 0. Any time you have this pairing, you can think of it as an association: to this thing, I pair up that thing. It's almost like, in some sense, being married right? Now, the question is: what do these relations look like? Let me give you another example: {(-1, 2),(1, 5),(3, 5)}. That's another example, where the connection is -1 here is being associated with 2, 1 here is being associated with 5, 3 here is being associated with 5. I'm constantly thinking about these things in terms of (x, y)--so I'm putting the x first and the y second--this is what I'm thinking about when I think of a relation.
Now, If you think about these things graphically, you can plot these things. So if we plot the first one, I see (3, -1). So there will be a point right there. (2, 4), it'll be way up here, and (3, 0) will be right here. So there's a graph of this relation. You can see all the three points there. What about here? Here I can graph this relation: (-1, 2), (1, 5), and (3, 5). There's a visualization of them. But it really is a connection between the first person with the second person.
Now, there's an extremely important, special example of relations, and those are called functions. And a function is nothing more than a relation where the first term, that number will only appear once in the list. Now, for example, if you look here, I see that this thing has a first term of 3, and this one has a first term of 3. So the first term actually appears twice. So therefore, this is not an example of a function. A function is a relation--so it's a collection of these ordered pairs--where the first term only appears once. So in this visual example, you can see, what does it mean for the first person to appear more than once? It means that there has to be two points on top of each other somehow. There has to be two points where they have the same first coordinate. In that case, we say this is not a function. What about this: {(-1, 2),(1, 5),(3, 5)}? Well here, I see that for every point, there is no other point that is on the same vertical line as the other ones. So therefore, this is a function. And functions are in fact what we're going to really, for the most part, study in algebra and in further areas of math. And you can see that something is not a function if it fails this vertical line test. So this vertical line test just means that every time a draw a vertical line, only at most one point will pass through that line. Here-- {(3, -1), (2, 4), (3, 0)}--it fails the vertical line test because a vertical line will pass through two points here--not a function. Here--{(-1, 2),(1, 5),(3, 5)}--every single one only passes through one point. Notice by the way that horizontally, these two share the same second coordinate (5), but that's not what's required to be a function, only that all the first coordinates are different. That's all that's involved.
Now, let me show you some examples of things that aren't necessarily numbers. You don't have to have numbers here. Here's a pairing, here's a connection: I have states paired up with institutions. So these are different institutions, these are in fact academic institutions. You can imagine penal institutions, where people go to prison, which in fact some of you may think some of these schools are. So Texas, for example, I associate with the University of Texas in Austin; Massachusetts I associate with Williams College; New York I associate with Columbia; Texas I associate with Rice University; California with UCLA; California with Harvey Mudd; and Ohio with Ohio State. So there's a state and here's a school from it. The question is: is this a function? Well, let's see. Does everything here appear at most once? Well, no. You can see for example that California actually appears twice. So for California we have (California, UCLA), we have (California, Harvey Mudd), so this would not be a function. Also notice that it fails because of Texas; I have (Texas, University of Texas), and (Texas, Rice University). You could draw a little picture like this, by the way, if you wanted to. Some people do this. Here's the thing of states, and here's the schools. And so here's Texas, and send an arrow to the school UT. And then here's MA (Massachusetts), and I send that to Williams. Here's New York--this is just a different way of representing this information, it's a bit more visual way--send that to Columbia. Here's Texas again, and now it's Rice, different school. Here's California; UCLA. And then California again also goes to Harvey Mudd. And then Ohio, that goes to Ohio State. And to see if this relation is really a function or not, the question is: for each point here, is there only one arrow coming out? But since I see here there are two arrows, you can see that this as a first coordinate is associated with two different people as second coordinates, both UT and Rice. Similarly, California actually has two things that it's paired up with. So if I say I'm thinking of Texas, you don't know for sure which school I'm thinking about, because there are two possibilities. This is not a function. It's not well-defined, we say. It's a relation, but it's not a function.
One last example; how about this thing? How about institutions and relate them with their mascots? So you have the University of Texas, which are the Longhorns; University of Tennessee, which are the Volunteers; Williams College--they're actually the Ephs, but let's think of them as the Purple Cows; University of Colorado are the Buffalo; and University of Washington are the Huskies. Well notice that this in fact is a function because each one of these schools is associated with only one thing here. And there's no "University of Texas has the Longhorns and the Beavers." There are no beavers at the University of Texas. So in fact, this is an example of a function. If you drew that little chart--in fact I'll just draw it over here, take a look at this, there it is, there's the chart right there. And you can see that out of every university there's only one arrow emanating going to the mascot. So this relation is actually an example of a function.
So there you have the connection between the special idea of a function, which is an example of the more general issue of a relation. And we'll take a look at functions in great, great detail, and we'll even see some relations along the way. So there's sort of just a general, global sense of what a relation really is about. A little abstract, but it'll be more concrete as you think about it. Good luck, and hook `em Horns!
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Introducing Relations and Functions Page [1 of 2]