discrete mathematics - Equality of Cardinality of Cartesian Sets

I'm having a hard time thinking about this problem.

My question: Let A be a set. Define C to be the collection of all functions f: {0,1} --> A. Prove that |A x A| = |C| by constructing a bijection F: A x A --> C.

I'm assuming A x A has the same cardinality as A itself. Also, to create a bijection from A x A --> C, I think I need to prove |A x A|≤|C| and |A x A|≥|C| through Cantor Schroder-Bernstein Theorem.

Can someone please tell me how to solve this?