Suppose that $G$ is a $d$-regular bipartite graph with bipartition $(A, B)$, and that $d ≥ 1$.
a) Prove that $∣A∣ = ∣B∣$.
b) Prove that $G$ contains a perfect matching.
So I understand that a $d$-regular graph will have every vertex of degree $d$. But I am not sure how that would apply here. Will I have to insert $d$ into my use of Hall's condition and then move from there. And also, how exactly would I prove that $|A| = |B|$? Would that also involve Hall's condition?
Thank you!