Refer to the definition of symmetric difference given above. Prove each of 47-52, assuming that A, B, and C are all subsets of a universal set U.
A Δ ∅ = A

To prove:

AΔϕ=A

Given information:

A be any sets.

Concept used:

AΔ∅=(A−∅)∪(∅−A)

Calculation:

The objective is to prove the following statement.

AΔ∅=A

The symmetric difference of A and B is defined as,

AΔB=(A−B)∪(B−A)

Consider the left hand side part.

AΔ∅=(A−∅)∪(∅−A)     By the definition of Δ.=(A∩∅)∪(∅∩Ac)  By the set difference law