   Chapter 1.2, Problem 3E

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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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