how do you prove a subSET is a subGROUP?
1) you verfiy the subset satisfies the group axioms:
a) closure (if a,b are in a subset S of G, we need a*b to be in S as well).
b) associativity (but this is obvious...why?)
c) existence of an identity (hint: doesn't this have to be the same identity as the one for G?)
d) existence of inverses
to save some time: prove that (a) & (d) together imply (c). this (together with (b) being obvious), only gives you two things to check, instead of four.
2) alternative method: show that for all a,b in S, a*b-1 is also in S. why is this just as good as doing (1)?
(hint: consider a = b, first, to show that e is in S. next, consider a = e, to show that b-1 is in S. finally, use the fact that b = (b-1)-1, to show that ab is in S).
to continue, observe that x in A⋂B, means: x is in A, AND x is in B.
to answer the next part, you need to know that |A⋂B| divides |A| and |B| (lagrange's theorem). what is gcd(10,7)?
for the last part: consider 2Z U 3Z. is 2+3 = 5 in this?