i cannot solve this exerccise

Question

Let M = {1, 2, 3, 4, 5, 6, 7, 8, 9}. As usual, denote ¨ P (M): = {A | A ⊆ M} is the power set of M. We consider the maps

⊕: P (M) × P (M) → P (M) and *: F2 × P (M) → P (M),

by

A ⊕ B: = A triangle B and α* A: = (∅ if α = [0], A if α = [1]),

for all ¨ A, B ∈ P (M) are given. Show that (P (M), ⊕,) is a vector space over F2.

Answer 1

Here you are!