Suppose that g is a function from A to B and f is a function from B to C. Prove each of these statements. a)If f◦g is onto, then f must also be onto. b)If f◦g is one-to-one, then g must also be one-to-one. c)If f◦g is a bijection, then g is onto if and only if f is one-to-one.
Suppose that g is a function from A to B and f is a function from B to C. Prove each of these statements. a)If f◦g is onto, then f must also be onto. b)If f◦g is one-to-one, then g must also be one-to-one. c)If f◦g is a bijection, then g is onto if and only if f is one-to-one.
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter1: Fundamentals
Section1.3: Properties Of Composite Mappings (optional)
Problem 12E: Let f:AB and g:BA. Prove that f is one-to-one and onto if fg is one to-one and gf onto.
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Suppose that g is a function from A to B and f is a function from B to C. Prove each of these statements.
a)If f◦g is onto, then f must also be onto.
b)If f◦g is one-to-one, then g must also be one-to-one.
c)If f◦g is a bijection, then g is onto if and only if f is one-to-one.
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