Let A, B and C be nonempty sets and let f : A → B, g: BC and h: B → C be functions.For each of the following, prove or disprove:(a) If f is bijective and gof= hof, then g = h.(b) If gof= hof, then g = h.

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Answered: Let A, B and C be nonempty sets and let…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter1: Fundamentals

Section1.2: Mappings

Problem 23E: Let a and b be constant integers with a0, and let the mapping f:ZZ be defined by f(x)=ax+b. Prove...

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2. In each of the following parts, a relation is defined on the set of all integers. Determine in each case whether or not is reflexive, symmetric or transitive. Justify your answers. a. if and only if . b. if and only if . c. if and only if for some in . d. if and only if . e. if and only if . f. if and only if . g. if and only if . h. if and only if . i. if and only if . j. if and only if . k. if and only if .
Label each of the following statements as either true or false. Let f:AB. Then f(A)=B for all nonempty sets A and B.
Let f:AB and g:BA. Prove that f is one-to-one and onto if fg is one to-one and gf onto.
Let a and b be constant integers with a0, and let the mapping f:ZZ be defined by f(x)=ax+b. Prove that f is one-to-one. Prove that f is onto if and only if a=1 or a=1.
21. A relation on a nonempty set is called irreflexive if for all. Which of the relations in Exercise 2 are irreflexive? 2. In each of the following parts, a relation is defined on the set of all integers. Determine in each case whether or not is reflexive, symmetric, or transitive. Justify your answers. a. if and only if b. if and only if c. if and only if for some in . d. if and only if e. if and only if f. if and only if g. if and only if h. if and only if i. if and only if j. if and only if. k. if and only if.
Label each of the following statements as either true or false. Let R be a relation on a nonempty set A that is symmetric and transitive. Since R is symmetric xRy implies yRx. Since R is transitive xRy and yRx implies xRx. Hence R is alsoreflexive and thus an equivalence relation on A.

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Let A, B and C be nonempty sets and let f : A → B, g: BC and h: B → C be functions. For each of the following, prove or disprove: (a) If f is bijective and go f = hof, then g = h. (b) If gof=hof, then g = h.