Proving Set Equality Using Algebraic Manipulation

[dlemke]

Prove C = G

C = {x + 7: x ∈ N} and G = {x : x ∈ N and x > 7}

I can come up with a logical answer to this, but I can't come up with an algebraic answer, and I'm not sure if that matters, I just want to see if there is a more proper way to do this than what I came up with, which is:

Let m ∈ N and x ∈ N, then m = x + 7 is such that m > 7 which is the definition of G. Because they share the same definition C ⊆ G and G ⊆ C and C = G.

And here's another one, same situation, I can do it logically but not algebraically:
{3x : x ∈ N} ∩ {x : x ∈ N} = {3x + 21 : x ∈ N}

 

[╔(σ_σ)╝]

It seems to me like you can use induction :-).