   Chapter 1.2, Problem 16E

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# Let g :   Z → Z be given by g ( x )   =   { x if  x     is even x + 1 2 if  x     is odd .         a. For S = { 3 ,   4 } , find g ( S ) and g − 1 ( g ( S ) ) .b. For   T = { 5 , 6 } , find g − 1 ( T ) and g ( g − 1 ( T ) ) .

(a)

To determine

g(S) and g1(g(S)) for S={3,4}, where g: be given by

g(x)={xifxisevenx+12ifxisodd

Explanation

Given information:

g: is given by g(x)={xifxisevenx+12ifxisodd

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}

Explanation:

Let S={3,4}

The number 3 is an odd integer and 4 is an even integer.

By using the given mapping, g(x)={xifxisevenx+12ifxisodd

g(3)=3+12=42=2 and

g(4)=4

Thus, g(S)=g({3,4})={2,4}

(b)

To determine

g1(T) and g(g1(T)) for T={5,6}, where g: be given by

g(x)={xifxisevenx+12ifxisodd

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