Does a Cartesian product have the distributive property with respect to a symmetric difference?

Question

Mathematics

Hello!
Please tell me if the following statement is true:
A x (B ∆ C) = (A x B) ∆ (A x C)

Answer 1

a) Suppose that (x, y) ∈ A x (B ∆ C). Then x ∈ A and y ∈ B ∆ C.
Let y ∈ B \ C. Then (x, y) ∈ (A x B) \ (A x C).
Let y ∈ C \ B. Then (x, y) ∈ (A x C) \ (A x B).
Therefore, (x, y) ∈ (A x B) Δ (A x C).
A x (B ∆ C) ⊂ (A x B) ∆ (A x C)
b) Assume that (x, y) ∈ (A x B) ∆ (A x C). Then x ∈ A and y ∈ B \ C, or y ∈ B \ C.
Therefore, y ∈ B ∆ C, and (x, y) ∈ A x (B ∆ C).
(A x B) Δ (A x C) ⊂ A x (B Δ C)
c) A x (B Δ C) ⊂ (A x B) Δ (A x C) and (A x B) Δ (A x C) ⊂ A x (B ∆ C) ⇒ A x (B ∆ C) = (A x B) ∆ (A x C) (by definition of equality).

Answer 2

Obviously yes.