Set Theory Please help!

[MattF]

I have a problem that wants me to prove that T is a subset of P. I really don't even know where to begin on this problem please help point me in the right direction. thank you.

Problem: Let P={(a,b,c):a,b,c, E Z, and a^2+b^2=c^2}, and T={(p,q,r): p=x^2-y^2, q=2xy, and r=x^2+y^2, wehre x,y E Z} Show that T is a subset of P.

What I know: If T is a subset of P that means that for any value k that is an Element of T it has to be an Element of P. but how to prove this I do not know.

 

[MattF]

Consider an arbitrary triplet in T, (g, h, i). We must show that (g, h, i) is in P using only the properties of an element of T.

(g, h, p) is in T.

So there exist x, y in Z such that g = x²- y², h = 2xy, and i = x² + y² by the definition of T.

Because Z is closed under multiplication, subtraction, and addition, x² - y², 2xy, and x² + y² are all in Z.

(1) So g, h, and i are all in Z

(2) g² + h² = (x² - y²)² + (2xy)² = x⁴ - 2x²y² + y⁴ + 4x²y² = x⁴ + 2x²y² + y⁴ = (x² + y²)² = i².

In short, g, h, and i are in Z and g²+ h²= i².

So (g, h, i) is in P.

But (g, h, i) was an arbitrary triplet in T.

So every triplet in T is in P.

So T is a subset of P.

This may not be the most elegant way to do this. But I think it is intuitive.

Thank you!