Q5 (i) Let f: N→ N² be the function given by f(n) = (1, n²).

Prove that f is injective.

What can you deduce about the relationship between the

cardinalities of N and N²?

(ii) Let p and q be integers. Prove that if 2^P = 3^q, then p = q = 0.

[Hint: consider separately all the possible cases of p and/or q being

positive/negative/zero.]

(iii) Let g: N² → N be the function given by g(m, n) = 2m3n.

Assuming the result from part (ii), or otherwise, prove that g is injective.

What can you deduce about the relationship between the cardinalities of N and N²?