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Elements Of Modern Algebra

8th Edition
Gilbert + 2 others
ISBN: 9781285463230

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Elements Of Modern Algebra

8th Edition
Gilbert + 2 others
ISBN: 9781285463230
Textbook Problem
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For each of the following mappings, write out f ( S ) and f 1 ( T ) for the given S and T , where f : Z Z .

a. f ( x ) = | x | ; S = Z E , T = { 1 , 3 , 4 }

b. f ( x ) = { x + 1 if  x is even x if  x is odd; S = { 0 , 1 , 5 , 9 } T = Z E

c. f ( x ) = x 2 ; S = { 2 , 1 , 0 , 1 , 2 } , T = { 2 , 7 , 11 }

d. f ( x ) = | x | x ; S = T = { 7 , 1 , 0 , 2 , 4 } .

(a)

To determine

f(S) and f1(T) for the given S and T, where f: defined by f(x)=|x|;S=E,T={1,3,4}

Explanation

Given information:

f(x)=|x|;S=E,T={1,3,4}

Formula used:

1) If f:AB and SA, then

f(S)={y|yB and y=f(x) for some xS}

The set f(S) is called the image of S under f.

2) For any subset T of B, the inverse image of T denoted by f1(T) and is defined by

f1(T)={x|xA and f(x)T}

3) For any real number x, the absolute value or modulus of x is denoted by |x| and is defined as,

|x|={x,ifx0x,ifx<0

Explanation:

The set S=E={±1,±3,±5,±7,} is a set of odd integers.

As |k|=k,|k|=k for any real number

Let k be any positive odd integer,

f(k)=|k|=k;kE

Let k be any negative odd integer,

f(k)=|k|=k;kE

Hence, f(S)=f(E)=f({±1,±3,±5,±7,})

={1,3,5,7,} is a set of positive odd integers

(b)

To determine

f(S) and f1(T) for the given S and T, where f: defined by f(x)={x+1ifxisevenxifxisodd,S={0,1,5,9},T=E

(c)

To determine

f(S) and f1(T) for the given S and T, where f: defined by f(x)=x2,S={2,1,0,1,2},T={2,7,11}

(d)

To determine

f(S) and f1(T) for the given S and T, where f: defined by f(x)=|x|x,S=T={7,1,0,2,4}

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