is a set isomorphic to the product space of it's partition?

Question

Show that if ${\{B,C\}}$ is a partition of an arbitrary set $A$ then the set of all functions from $A$ to $\mathbb{R}$ - $\mathbb{R}^A$ is isomorphic to $\mathbb{R}^B\times\mathbb{R}^C$

I don't know how to approach this, can anyone help?

Answer 1

I assume that you are looking at $\Bbb R^D$ as a group under pointwise addition of maps, and $\Bbb R^E \times \Bbb R^F$ as the direct product of the two groups.

Hint: define $\Phi: \Bbb R^A \to \Bbb R^B \times \Bbb R^C$ by $\Phi(f) = (f\mid_B, f\mid_C)$. $\Phi$ is a bijective group homomorphism, so $\Bbb R^A \cong \Bbb R^B \times \Bbb R^C$.