SUNY Geneseo Department of Mathematics
Wednesday, April 18
Math 239 01
Spring 2018
Prof. Doug Baldwin
The question from the end of last class: can two functions be the same if they have different codomains.
Sundstrom: codomains have to be equal.
Set theory: A function is a set of ordered pairs, which implicitly defines the domain (the set of values appearing as the first coordinate of some pair), the range (the set of values appearing as the second coordinate of some pair), and the mapping, but codomains aren’t part of this definition at all.
If the exact definition of equality is a issue in this course we’ll use Sundstrom’s definition.
Section 6.5.
Find inverses of the following functions from ℝ to ℝ:
Notice that this isn’t a function, first because it defines 2 images for each y value, and second because it is undefined for much of its supposed domain (ℝ). Also notice that we can’t sweep away the two images problem by just assuming we only want the positive square root of y-1, because the definition of “inverse” requires that f-1 contain a pair for every pair in f. For example, f contains (-2,5) and (2,5), and thus f-1 must contain (5,-2) and (5,2).
This time f-1 is a function.
Define the notation “x mod y,” for integers x and y, to mean the integer k such that x = qy + k for some integer q and 0 ≤ k < y. By Corollary 3.32, k is unique for any given x and y.
Suppose f : {0,1,...,999} → {0,1,...,999} is defined by f(n) = (n+100) mod 1000.
Also suppose g : {0,1,...,999} → {0,1,...,999} is defined by g(n) = (n+900) mod 1000.
Prove that g = f-1.
Idea: show that for all n, g(f(n)) = n and f(g(n)) = n. This is a method you may have learned in an algebra class, but is it in fact justified by what you know about inverses now? It turns out it is, specifically by the definition of “inverse.” The reasoning is that if g(f(n)) = n then there must be some m such that f(n) = m, i.e., the pair (n,m) is in f, and if g(m) = n we further know that (m,n) is in g, so every ordered pair in f has its reverse in g. By similar reasoning, if f(g(n)) = n then (n,m) is in g for some m and (m,n) is in f, so the only pairs in g are reversals of ones in g. Thus g satisfies the definition of f-1.
Sketch out the main proof that g = f-1 for Friday, or identify questions that need to be answered about the proof.
The definition of “inverse.”
Inverses may be functions but don’t have to be.
Thinking mathematically, e.g., about how fundamental definitions justify common methods for doing something.
Finish the proof above.
Start talking about relations.