Question

#1: a) Let a, b, c be integers and x is congruent to y (mod m). Prove that ax^2+bx+c is congruent to ay^2+by+c (mod m).

b) Show that the result in part a may be false if a, b, c are not all integers, even if ax^2+bx+c and ay^2+by+c are both integers.

#2: Let p: X --> Y and q: Y -->Z be functions such that (q(p)): X-->Z is onto. Prove that q is onto and give an example to show that p need not be onto.

#3: Give an algebraic proof to show that nCr = (n-1) C (r-1) + (n-1)C r.

Hi! Can some please help me with these problems? They are a little bit confusing and I am stuck on them.