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Elements Of Modern Algebra

8th Edition
Gilbert + 2 others
ISBN: 9781285463230

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BuyFindarrow_forward

Elements Of Modern Algebra

8th Edition
Gilbert + 2 others
ISBN: 9781285463230
Textbook Problem
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For each of the following mappings f : Z Z , exhibit a right inverse of f with respect to mapping composition whenever one exists.

a. f ( x ) = 2 x b. f ( x ) = 3 x

c. f ( x ) = x + 2 d. f ( x ) = 1 x

e. f ( x ) = x 3 f. f ( x ) = x 2

g. f ( x ) = { x if  x  is even 2 x 1 if  x  is odd h. f ( x ) = { x if  x  is even x 1 if  x  is odd

i. f ( x ) = | x | j. f ( x ) = x | x |

k. f ( x ) = { x if  x  is even x 1 2 if  x  is odd l. f ( x ) = { x 1 if  x  is even 2 x if  x  is odd

m. f ( x ) = { x 2 if  x  is even x + 2 if  x  is odd n. f ( x ) = { x + 1 if  x  is even x + 1 2 if  x  is odd

(a)

To determine

For the following mapping f:, exhibit a right inverse of f with respect to mapping composition whenever one exists.

Explanation

Given Information:

f(x)=2x

Formula used:

1) A standard way to demonstrate that f:AB is onto is to take an arbitrary element b in B and show that there exists an element aA such that b=f(x).

2) To show that a given mapping f:AB is not onto, find single element b in B for which no xA exist such that b=f(x).

3) Let A be a nonempty set, and f:AA. Then f is an onto mapping if and only if f has a right inverse

(b)

To determine

For the following mapping f:, exhibit a right inverse of f with respect to mapping composition whenever one exists.

(c)

To determine

For the following mapping f:, exhibit a right inverse of f with respect to mapping composition whenever one exists.

(d)

To determine

For the following mapping f:, exhibit a right inverse of f with respect to mapping composition whenever one exists.

(e)

To determine

For the following mapping f:, exhibit a right inverse of f with respect to mapping composition whenever one exists.

(f)

To determine

For the following mapping f:, exhibit a right inverse of f with respect to mapping composition whenever one exists.

(g)

To determine

For the following mapping f:, exhibit a right inverse of f with respect to mapping composition whenever one exists.

(h)

To determine

For the following mapping f:, exhibit a right inverse of f with respect to mapping composition whenever one exists.

(i)

To determine

For the following mapping f:, exhibit a right inverse of f with respect to mapping composition whenever one exists.

(j)

To determine

For the following mapping f:, exhibit a right inverse of f with respect to mapping composition whenever one exists.

(k)

To determine

For the following mapping f:, exhibit a right inverse of f with respect to mapping composition whenever one exists.

(l)

To determine

For the following mapping f:, exhibit a right inverse of f with respect to mapping composition whenever one exists.

(m)

To determine

For the following mapping f:, exhibit a right inverse of f with respect to mapping composition whenever one exists.

(n)

To determine

For the following mappings f:, exhibit a right inverse of f with respect to mapping composition whenever one exists.

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