Consider the following two sets A & B: A= {4, 8, 12, 16, … } B = {1, 3, 5, 7, 9, … } Let
be a function from z × z to , such that f(m,n) = (m*m)-(n*n). . i) Show that every element of the set A has a preimage under the function f. Type/Insert your answer here! ii) Show that every element of the set B has a preimage under the function f. Type/Insert your answer here!

Given:

Consider the following two sets A & B:

A= {4, 8, 12, 16, … }

B ={1, 3, 5, 7, 9, … }

Let f be a function from z × z to , such that $f(m,n) = (m*m)-(n*n)$ .

1) Show that every element of the set A has a preimage under the function f.

Solution:

$\forall a\in A=>a=4*l,l\in N\\ f(l+1,l-1)=(l+1)^2-(l-1)^2=4*l=a$

2)Show that every element of the set B has a preimage under the function f. 

Solution:

$\forall b\in B=>b=2*l+1,l\in N or 0\\ f(l+1,l)=(l+1)^2-l^2=2*l+1=b$

Answer:\\

(1) for$\ a\in A \ preimage (\frac{a}{4}+1,\frac{a}{4}-1)$

(2) for$\ b\in B \ preimage (\frac{b-1}{2}+1,\frac{b-1}{2})$