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Chapter 1: Problem 20

Let \(f: A \rightarrow B\) and \(g: B \rightarrow C\) be functions. (a) Show that if \(g \circ f\) is injective, then \(f\) is injective. (b) Show that if \(g \circ f\) is surjective, then \(g\) is surjective.

Short Answer

Expert verified
If \(g \circ f\) is injective, then \(f\) is injective because \(f\) has to assign a unique element from set \(A\) to a unique element in set \(B\) such that there are no other members in \(A\), which maps to the same element in \(B\). If \(g \circ f\) is surjective, then \(g\) is surjective because for every element in set \(C\), there must be an element in set \(B\) for which the function \(g\) maps to this element of set \(C\).

Step by step solution

01

Understanding the Context

The notation \(g \circ f\) represents a composition of functions, where the output of function \(f\) serves as the input for function \(g\). By applying definitions and properties of injective and surjective functions, we can examine the conditions whereby if \(g \circ f\) is injective, then \(f\) is injective, and if \(g \circ f\) is surjective, then \(g\) is surjective.
02

Proving f is injective

Assume that \(g(f(a_1)) = g(f(a_2))\). If \(g \circ f\) is injective as per our assumption, it means that \(a_1\) must equal \(a_2\), proving that \(f\) is injective.
03

Proving g is surjective

If \(g \circ f\) is surjective, for every element \(c\) in \(C\) there is an element \(a\) in \(A\) such that \(g(f(a)) = c\). This implies there must exist an element in \(B\), which is \(f(a)\), such that \(g(f(a)) = c\). Thus, \(g\) is surjective.

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Most popular questions from this chapter

For each \(n \in \mathbb{N}\), let \(A_{n}=\\{(n+1) k: k \in \mathbb{N}]\). (a) What is \(A_{1} \cap A_{2} ?\) (b) Determine the sets \(\bigcup\left\\{A_{n}: n \in \mathbb{N}\right\\}\) and \(\bigcap\left(A_{n}: n \in \mathbb{N}\right)\).

Determine the number of elements in \(\mathcal{P}(S)\), the collection of all subsets of \(S\), for each of the following sets: (a) \(S:=\\{1,2\\}\) (b) \(\quad S:=(1,2,3)\), (c) \(S:=(1,2,3,4\\}\). Be sure to include the empty set and the set \(S\) itself in \(\mathcal{P}(S)\).

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Draw diagrams in the plane of the Cartesian products \(A \times B\) for the given sets \(A\) and \(B\). (a) \(A=\\{x \in \mathbb{R}: 1 \leq x \leq 2\) or \(3 \leq x \leq 4\\}, B=\\{x \in \mathbb{R}: x=1\) or \(x=2\\}\). (b) \(A=\\{1,2,3\\}, B=\\{x \in \mathbb{R}: 1 \leq x \leq 3\\}\).
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