Define a mapping fn : Z → Z by fn(x) = 1 – nx for each integer n E Z.a. For which values of n is fn onto?b. For which values of n is fn one-to-one?

Answered: Image /qna-images/answer/372558d5-a107-4338-a53d-8eedffc89729.jpg

Answered: Define a mapping fn : Z → Z by fn(x) =…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter1: Fundamentals

Section1.5: Permutations And Inverses

Problem 10E: 10. Let and be mappings from to. Prove that if is invertible, then is onto and is...

See similar textbooks

Similar questions

4. Let , where is nonempty. Prove that a has left inverse if and only if for every subset of .
27. Let , where and are nonempty. Prove that has the property that for every subset of if and only if is one-to-one. (Compare with Exercise 15 b.). 15. b. For the mapping , show that if , then .
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.
Let ab in a field F. Show that x+a and x+b are relatively prime in F[x].
10. Let and be mappings from to. Prove that if is invertible, then is onto and is one-to-one.
For an element x of an ordered integral domain D, the absolute value | x | is defined by | x |={ xifx0xif0x Prove that | x |=| x | for all xD. Prove that | x |x| x | for all xD. Prove that | xy |=| x || y | for all x,yD. Prove that | x+y || x |+| y | for all x,yD. Prove that | | x || y | || xy | for all x,yD.
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
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
Define powers of a permutation on by the following: and for Let and be permutations on a nonempty set . Prove that for all positive integers .
Let D be an integral domain with four elements, D=0,e,a,b, where e is the unity. a. Prove that D has characteristic 2. b. Construct an addition table for D.

Recommended textbooks for you

Elements Of Modern Algebra

Algebra

ISBN:

9781285463230

Author:

Gilbert, Linda, Jimmie

Publisher:

Cengage Learning,

Linear Algebra: A Modern Introduction

Algebra

ISBN:

9781285463247

Author:

David Poole

Publisher:

Cengage Learning

Elements Of Modern Algebra

Algebra

ISBN:

9781285463230

Author:

Gilbert, Linda, Jimmie

Publisher:

Cengage Learning,

Linear Algebra: A Modern Introduction

Algebra

ISBN:

9781285463247

Author:

David Poole

Publisher:

Cengage Learning

SEE MORE TEXTBOOKS

Define a mapping fn : Z → Z by fn(x) = 1 – nx for each integer n E Z. a. For which values of n is fn onto? b. For which values of n is fn one-to-one?