Explanation Given: The functions f and g are defined as f : X → Y and g : Y → Z respectively. And, the function h = g ∘ f is a one to one correspondence. Concept used: A map f : A → B is said to be one-to-one if an element of B does not have more than one pre-image in A. In other words, amap f : A → B is said to be one-to-one if f ( a ) = f ( b ) ⇒ a = b . Proof: Let x 1 , x 2 ∈ X , y 1 , y 2 ∈ Y and z 1 , z 2 ∈ Z such that f ( x 1 ) = y 1 , f ( x 2 ) = y 2 , g ( y 1 ) = z 1 , and g ( y 2 ) = z 2 . Also, h ( x 1 ) = z 1 and h ( x 2 ) = z 2 . Since h = g ∘ f and f : X → Y , g : Y → Z , the function h can be defined as h : X → Z To determine (b) To prove that the function g is onto.

3rd Edition

ISBN: 9781284056242

Author: David J. Hunter

Publisher: Jones & Bartlett Learning

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Chapter 2.3, Problem 35E

To determine

(a)

To prove that f is one to one.

To determine

(b)

To prove that the function g is onto.

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Chapter 2.3, Problem 34E

Chapter 2.3, Problem 36E

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(a) To prove that f is one to one.

Solution Summary: The author explains how the functions f and g are defined, and the function h is a one-to-one correspondence.