Consider the following functions and determine if they are bijective. [A function is said to be bijective or bijection, if a function f: A→B is both one-to-one and onto.] (a) f: Z × Z→Z, f(n, m) = n2 + m2 (b) f: R→R, f(x) = x3 − 3 (c) f: R × R→R, f(n, m) = 2m − n

One-to-one function is a function f that maps distinct elements to distinct elements.

Onto function is a function f that maps an element x to every element y.

a) The function is not onto, because $n^2+m^2\ge0$ ; and the function is not one-to-one, because

$f(n,m)=f(-n,-m)$

So, the function is not bijective.

b) The function is both one-to-one and onto.

So, the function is bijective.

c) The function is not one-to-one: for example: $f(3,2)=1=f(1,1)$

So, the function is not bijective.