   Chapter 1.2, Problem 13E

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# For the given f :   A   →   B , decide whether f is onto and whether it is one-to-one. Prove that yourdecisions are correct.a. Α     =     Z   ×   Z ,     B   =   Z × Z ,     f ( x , y )   =   ( y , x ) b. Α     =     Z   ×   Z ,     B   =   Z ,     f ( x , y )   =   x + y c. Α     =     Z   ×   Z ,     B   =   Z ,     f ( x , y )   =   x d. Α     =     Z   ,     B   =   Z × Z ,     f ( x )   =   ( x ,   1 ) e. Α     =     Z   × Z ,     B   =   Z ,     f ( x , y )   =   x 2 f. Α     =     Z   × Z ,     B   =   Z ,     f ( x , y )   =   x 2 + y 2 g. Α     =     Z + ×   Z + ,     B   =   Q ,     f ( x , y )   =   x / y h. Α   =     R × R ,   B   =   R ,   f ( x , y )   =   2 x + y

(a)

To determine

Whether the mapping f(x,y)=(y,x) is one-to-one and it is onto. Prove that the decisions are correct.

Explanation

Given information:

f:AB where A=×,B=×, f(x,y)=(y,x)

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.

Explanation:

Let x1=(a1,b1),x2=(a2,b2)× be any elements

(b)

To determine

Whether the mapping f(x,y)=x+y is one-to-one and it is onto. Prove that the decisions are correct.

(c)

To determine

Whether the mapping f(x,y)=x is one-to-one and it is onto. Prove that the decisions are correct.

(d)

To determine

Whether the mapping f(x)=(x,1) is one-to-one and it is onto. Prove that the decisions are correct.

(e)

To determine

Whether the mapping f(x,y)=x2 is one-to-one and it is onto. Prove that the decisions are correct.

(f)

To determine

Whether the mapping f(x,y)=x2+y2 is one-to-one and it is onto. Prove that the decisions are correct.

(g)

To determine

Whether the mapping f(x,y)=xy is one-to-one and it is onto. Prove that the decisions are correct.

(h)

To determine

Whether the mapping f(x,y)=2x+y is one-to-one and it is onto. Prove that the decisions are correct.

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