Cartesian Product

Last updated Nov 1, 2022

Let $A, B$ be Sets. The Cartesian Product, denoted $A \times B$ is defined as the Set $$A \times B := {(a, b) : a \in A, b \in B}$$

Consider $\mathcal{P}(\mathcal{P}(A \cup B))$. Observe that for $a \in A$ and $b \in B$, $$(a, b) = {{a}, {a, b}} \subset \mathcal{P}(A \cup B),$$ so $(a, b) \in \mathcal{P}(\mathcal{P}(A \cup B))$. Thus, by the Axiom Schema of Specification we can construct

$${(a, b) \subset \mathcal{P}(A \cup B) : a \in A \wedge b \in B } = {(a, b) : a \in A, b \in B} := A \times B.$$ $\blacksquare$