Question

4. Let A and B be sets. Prove the commutative laws from Table 1 by showing that

a) A ∪ B = B ∪ A.

b) A ∩ B = B ∩ A.

Answer 1

a) Let $x∈A∪B.$ Then $x∈A$ or $x∈B.$ Which implies  $x∈B$ or $x∈A.$

Hence $x∈B∪A.$

Thus $A∪B⊆B∪A.$

Similarly, we can show that $B∪A⊆A∪B.$

Therefore, $A∪B=B∪A.$

b) Let $x∈A∩B.$ Then $x∈A$ and $x∈B.$ Which implies $x∈B$ and $x∈A.$ Hence $x∈B∩A.$

Thus $A∩B⊆B∩A.$

Similarly, we can show that $B∩A⊆A∩B.$

Therefore, $A∩B=B∩A.$