Either provide an informal proof of OR a counterexample to each of the following. If g and h are relations on R x R, and g and g ∘ h are both one-to-one, then h is one-to-one. If f is a function from finite set A to finite set B and |A| = |B|, then f is one-to-one if and only if f is onto.

Either provide an informal proof of OR a counterexample to each of the following. If g and h are relations on R x R, and g and g ∘ h are both one-to-one, then h is one-to-one. If f is a function from finite set A to finite set B and |A| = |B|, then f is one-to-one if and only if f is onto.

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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Question

Either provide an informal proof of OR a counterexample to each of the following.

If g and h are relations on R x R, and g and g ∘ h are both one-to-one, then h is one-to-one.
If f is a function from finite set A to finite set B and |A| = |B|, then f is one-to-one if and only if f is onto.

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