Final answer:
The proof for a × (b ∪ c) ⊆ (a × b) ∪ (a × c) involves demonstrating that each element in a × (b ∪ c) is also in (a × b) ∪ (a × c) by considering the elements of the sets and their membership.
Step-by-step explanation:
Proof for the Distributive Property of Cartesian Product over Union
To prove that for any sets a, b, and c, a × (b ∪ c) ⊆ (a × b) ∪ (a × c), we need to show that every element in the set a × (b ∪ c) is also an element of the set (a × b) ∪ (a × c).
Let's take a generic element (x, y) where x ∈ a and y ∈ (b ∪ c). Since y is either in b or c, two cases arise: If y ∈ b, then (x, y) ∈ (a × b), and if y ∈ c, then (x, y) ∈ (a × c). In either case, (x, y) is in the union (a × b) ∪ (a × c), hence proving that a × (b ∪ c) ⊆ (a × b) ∪ (a × c).