Relations Problem 4


This time, the objects in our relation are functions. Don't let that worry you. But, when you write out the standard outline for an equivalence relation proof, you'll need to pick a function, or two functions, or three functions where you might normallly pick one, two, or three numbers.

What three properties is an equivalence relation suppose to have? Your proof should divide into three separate labelled sections, one for each propery.

Now look at the definition of the relation: there is a \(k \in \mathbb{R}\) such that f(x) = g(x) for every \(x \geq k\). This is almost the same as saying that the two functions are identical, except that we're ignoring their behavior on values below some bound k. Draw a graph of two functions that are the same on larger values but differ on smaller values, so you have a picture of what's going on.

Notice that you pick the pair of functions first, then choose k. So k could be different for different pairs of functions.