Suppose f: A --> B is a function and g is the powerset of B to the powerset of A, while g(S) = f-1(S) for all S subset of B. Prove or disprove that if g is one-to-one then f is onto, and if g is onto then f is one-to-one.

Suppose f: A --> B is a function and g is the powerset of B to the powerset of A, while g(S) = f-1(S) for all S subset of B. Prove or disprove that if g is one-to-one then f is onto, and if g is onto then 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 f: A --> B is a function and g is the powerset of B to the powerset of A, while g(S) = f^-1(S) for all S subset of B. Prove or disprove that if g is one-to-one then f is onto, and if g is onto then f is one-to-one.

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