Consider the set of integers Z. The set Z x Z consists of all ordered pairs of integers. In symbols, Z x Z={(x,y):x,y∈Z}
For example, (2,-5) , (0,0) , (-127,10) , and (-5,2) are all distinct elements of Z x Z.
Notice that the order matters; (2,-5) and (-5,2) are different elements of Z x Z.
Now let?s define a few functions on Z x Z:
f(x,y)=x*y
h(x,y)=x^y
(a) What is the range of each of these functions? In other words, is the image an integer, a rational number or only a real number?
(b) For which of these functions (if any) is the image of (x,y) equal to the image of (y,x)?