Question

A∩(BΔC) = (A∩B)Δ(A∩C)

Answer 1

Answer:

Prove

A∩(BΔC)=(A∩B)Δ(A∩C)

without using Venn Diagram

My Try:

we have

A∩(BΔC)=A∩((B∪C)−(B∩C))=A∩((B∪C)∩(B∩C)′)

⟹

A∩(BΔC)=(A∩(B∪C))∩(B′∪C′)

⟹

A∩(BΔC)=((A∩B)∪(A∩C))∩(B′∪C′)

⟹

A∩(BΔC)=((A∩B)∩(B′∪C′))∪((A∩C)∩(B′∪C′))

⟹

A∩(BΔC)=(A∩B∩C′)∪(A∩C∩B′)

⟹

A∩(BΔC)=((A∩B)−C)∪((A∩C)−B)