show that the function
`(i) f : N to N : f (x) =x^(2)` is one-one into
(ii) `f : Z to Z : f (x) =x^(2)` is many -one into .

Explanation

Let x,y `in` N and f (x) = f(y)
`rArr x^(2)=y^(2)`
`rArr x^(2)-y^(2)` = 0
`rArr (x-y)(x+y) = 0`
`rArr x-y =0` (`because x + y ne 0`)
`rArr` x=y
Therefore, f is one one.
Again , let f (x) = y, where y `in` N (co-domain)
`rArr x^(2) = y`
`rArrx=+-sqrt(y)in N is y -3 in N`
Therefore , an element 3 belongs to co-domain such that it has no pre-image in domain N.
`:.` f is into.
Therefore, f is one-one into.