Define powers of a permutation
Let
for all positive integers
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Chapter 2 Solutions
ELEMENTS OF MODERN ALGEBRA
- 6. Prove that if is a permutation on , then is a permutation on .arrow_forwardLet f1,f2,...,fn be permutations on a nonempty set A. Prove that (f1f2...fn)1=fn1=fn1...f21f11 for all positive integers n.arrow_forwardLet 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.arrow_forward
- 4. Let , where is nonempty. Prove that a has left inverse if and only if for every subset of .arrow_forwardExercises 13. For the given permutations, and , find a permutation such that is the conjugate of by –that is, such that . a. ; b. ; c. ; d. ; e. ; f. ;arrow_forwardLet f and g be permutations on A. Prove that (fg)1=g1f1.arrow_forward
- 30. Prove statement of Theorem : for all integers .arrow_forward31. Prove statement of Theorem : for all integers and .arrow_forward21. 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.arrow_forward
- Let a and b be integers such that ab and ba. Prove that b=0.arrow_forwardFor 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)={ 2x1ifxiseven2xifxisoddarrow_forwardLet f:AB and g:BA. Prove that f is one-to-one and onto if fg is one to-one and gf onto.arrow_forward
- Elements Of Modern AlgebraAlgebraISBN:9781285463230Author:Gilbert, Linda, JimmiePublisher:Cengage Learning,Elementary Linear Algebra (MindTap Course List)AlgebraISBN:9781305658004Author:Ron LarsonPublisher:Cengage LearningLinear Algebra: A Modern IntroductionAlgebraISBN:9781285463247Author:David PoolePublisher:Cengage Learning