Hello,
I need some help with this problem:
Let S = {x ∈: x^{2} ∈)
I need to come up with a bijection between S and Q.
Any help appreciated.
Thanks!
OK, for one thing the INVERSE of f (if it exists, which you have NOT shown) would be a function $\displaystyle g: \mathbb{Q} \to S$.
so you would need to show TWO things:
1) $\displaystyle f \circ g = 1_{\mathbb{Q}}$
2)$\displaystyle g \circ f = 1_S$
but you can't simply say "g exists" you have to DEFINE it.
Perhaps a more straight-forward method is to prove:
A) $\displaystyle f(s_1) = f(s_2) \implies s_1 = s_2 $ (that is, f is injective)
B) for EVERY $\displaystyle q \in \mathbb{Q}$, there exists SOME $\displaystyle s \in S$ with: $\displaystyle f(s) = q$ (that is, f is surjective).