   Chapter 1.2, Problem 4E

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# For each of the following mappings f :   Z   →   Z , determine whether the mapping is onto and whether it is one-to-one. Justify all negative answers.a. f ( x )     =   2 x   b. f ( x )     =   3 x   c. f ( x )     =     x   + 3   d. f ( x )     =   x 3 e. f ( x )     =   | x | f. f ( x )     =   x   −     |   x   | 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   if  x     is even x   −   1 2   if  x     is odd j. f ( x )   =   { x   −     1 if  x     is even 2 x   if  x     is odd

(a)

To determine

Whether the mapping f(x)=2x is one-to-one and it is onto, and justify all negative answers.

Explanation

Given information:

f: is defined by 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) To show that f is not one-to-one, find two elements a1A and a2A such that a1a2 and f(a1)=f(a2).

4) To show that f is one-to-one by assuming that f(a1)=f(a2) and proving that this implies that a1=a2

(b)

To determine

Whether the mapping f(x)=3x is one-to-one and it is onto, and justify all negative answers.

(c)

To determine

Whether the mapping f(x)=x+3 is one-to-one and it is onto, and justify all negative answers.

(d)

To determine

Whether the mapping f(x)=x3 is one-to-one and it is onto, and justify all negative answers.

(e)

To determine

Whether the mapping f(x)=|x| is one-to-one and it is onto, and justify all negative answers.

(f)

To determine

Whether the mapping f(x)=x|x| is one-to-one and it is onto, and justify all negative answers.

(g)

To determine

Whether the mapping f(x)={xifxiseven2x1ifxisodd is one-to-one and it is onto, and justify all negative answers.

(h)

To determine

Whether the mapping f(x)={xifxisevenx1ifxisodd is one-to-one and it is onto, and justify all negative answers.

(i)

To determine

Whether the mapping f(x)={xifxisevenx12ifxisodd is one-to-one and it is onto, and justify all negative answers.

(j)

To determine

Whether the mapping f(x)={x1ifxiseven2xifxisodd is one-to-one and it is onto, and justify all negative answers.

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