Composition of functions
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This document discusses composition of functions and onto functions. It provides examples and proofs of the following statements: 1) The composition of onto functions is onto. It proves this by showing that for any element in the target of the second function g, there is an element in the domain of the composition g ◦ f that maps to it. 2) The converse is not true - if g ◦ f is onto, f and g individually do not have to be onto. It provides a counterexample. 3) If g ◦ f is onto, then g must be onto. It proves this by contradiction, considering when the intermediate function f is and is not onto.
Composition of functions
- 1. Working with composition of functions and onto. Tyler Murphy March 12, 2014 Working with composition of functions and onto. Let’s consider an example problem that was in the book. Problem 3.2.24. 3.2.24 (a) Prove that the composition of onto functions is onto. (b) Show, by example, that the converse of (a) is not true. (c) Show that if g ◦ f is onto, then g must be onto. 3.2.24.a (a) Prove that the composition of onto functions is onto. Proof. Let f and g be onto functions, and A, B, C are sets. Define f : A → B and g : B → C. WTS: g ◦ f is onto. This means we want to show that for every element in the target of g, there is an element in the domain of g ◦ f that goes to it. So we want to show that for any element, say z ∈ C, there is an element, say x ∈ A that get moved over to it. Let z ∈ C. Since g is onto, then every element in C can be written to as g(y) for some element y in B. So, z = g(y). Since f is onto, then every element in B can be written as f(x) for some element x ∈ A. So y = f(x). By subsitution, we have that z = g(f(x)) which is (g ◦ f)(x) = z. So, every element in C gets hit by some element in A by g ◦ f. So, g ◦ f is onto. 1
- 2. 3.2.24.b Here, we just want to find a counterexample to the claim that if g ◦ f is onto, then f is onto AND g is onto. Consider the sets: A = {0, 1}. B = {3, 4, 5}. C = {6, 7}. Define g ◦ f = {(0, 6), (1, 7)}, which is onto. Define g = {(3, 6), (4, 6), (5, 7)} is onto. But f = {(0, 3), (1, 4)} is not onto. BAM. DONE. 3.2.24.c Prove that if g ◦ f is onto, then g must be onto. First, let’s consider some incorrect thought processes. Define some sets A, B, C. A = {0, 1, 2}. B = {3, 4}. C = {6, 7, 8}. Define f : A → B as f = {(0, 3), (1, 3), (2, 4)}. Note that f is onto. Define g : B → C as g = {(3, 6), (4, 7)}. Note that g is not onto. Now here’s where you have to be careful. You cannot define g◦f as {(0, 6), (1, 7), (2, 8)}. Now, this set does represent some onto function from A → C, but this function is not g ◦f. Consider what it means to be in g ◦ f. This is g(f(x)) for some x in the dom(f). So g ◦ f is {(x, g(f(x))) | x ∈ A. So g ◦ f = {(0, g(f(0)), (1, g(f(1)), (2, g(f(2))}. So, g ◦ f is {(0, g(3)), (1, g(3)), (2, g(4))} So, g ◦ f is {(0, 6), (1, 6), (2, 7)}. Now you can see that g◦f is not onto Be careful when you are composing your functions that you don’t just go from the domain of f directly to the range of g. You have to stop in the middle at the ran(f) which is the dom(g). Now, let’s get to the proof. Claim: if g ◦ f is onto, then g must be onto. Proof. Let f and g be functions and g ◦ f be an onto function. Let A, B, C be sets such that f : A → B and g : B → C. Now, since g ◦ f is onto, we know that every element in C gets hit by some element in f mapped through g. Suppose g is not onto. Then there is some element, say z ∈ C such that g(y) = z, y ∈ B. Now we need to consider what happens when f is onto and when f is not onto. 2
- 3. Case1: Suppose f is onto. If f is onto, then we have every element in B can be written as f(x) for some x ∈ A. So y = f(x). So, g(f(x) = z. So g ◦ f is not onto, a contradiction. So, if f is onto, and g ◦ f is onto, then g must be onto. Case2: Suppose f is not onto. So there is some element in B, say y, such that f(x) = y for any element x ∈ A. Since g is not onto, then there is some element, say z ∈ C such that g(y) = z, y ∈ B. So, then for some element x ∈ A, f(x) = y and g(y) = z, so not everything in C gets hit. So g ◦ f is not onto, a contradiction. So, If g ◦ f is onto, then g must be onto. 3