Let's break down and solve the problem step by step. The image contains two parts of a mathematical proof related to set theory and cardinality.

Part 1a:

We are asked to prove the following equivalences related to sets: $(B \times C)^A \sim B^A \times C^A \quad \text{and} \quad (C^B)^A \sim C^{A \times B}$ where $B^A = \{ f \mid f: A \to B \}$, and $\sim$ denotes equinumerosity (sets having the same cardinality).

Step-by-step solution:

1. First equivalence: $(B \times C)^A \sim B^A \times C^A$

   Consider the meaning of $(B \times C)^A$. By definition, $(B \times C)^A$ is the set of all functions from $A$ to $B \times C$. A function $f: A \to (B \times C)$ assigns to each element $a \in A$ a pair $(b_a, c_a) \in B \times C$. In other words, it can be thought of as two functions:

   • $f_1: A \to B$