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Elements Of Modern Algebra

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter1: Fundamentals

Section1.5: Permutations And Inverses

Problem 4E: 4. Let , where is nonempty. Prove that a has left inverse if and only if for every subset of .

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2. Give an example of onto but not 1-1 function f: Z>0 → Z. (In your answer you
should also justify that f is a well-defined function when necessary, that is f(x) is
unique for each x = Q.)

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Let f:AA, where A is nonempty. Prove that f a has right inverse if and only if f(f1(T))=T for every subset T of A.
4. Let , where is nonempty. Prove that a has left inverse if and only if for every subset of .
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.
Prove that if f is a permutation on A, then (f1)1=f.
For each of the following parts, give an example of a mapping from E to E that satisfies the given conditions. a. one-to-one and onto b. one-to-one and not onto c. onto and not one-to-one d. not one-to-one and not onto
18. Let and be defined as follows. In each case, compute for arbitrary . a. b. c. d. e.
For the given f:ZZ, decide whether f is onto and whether it is one-to-one. Prove that your decisions are correct. a. f(x)={ x2ifxiseven0ifxisodd b. f(x)={ 0ifxiseven2xifxisodd c. f(x)={ 2x+1ifxisevenx+12ifxisodd d. f(x)={ x2ifxisevenx32ifxisodd e. f(x)={ 3xifxiseven2xifxisodd f. f(x)={ 2x1ifxiseven2xifxisodd
(See exercise 24.) Show that the relation f(x)Rg(x) if and only if f(x)=g(x) is an equivalence relation on R[ x ]. Describe the equivalence class [ f(x) ]. For each f(x)=i=0naixi in R[ x ], the formal derivative of f(x) is the polynomial f(x)=i=1niaixi1. (For n=0, f(x)=0 by definition.) Prove that [ f(x)+g(x) ]=f(x)+g(x). Prove that [ f(x)g(x) ]=f(x)g(x)+f(x)g(x).

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