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 B ) Δ C = A Δ ( B Δ C )

To prove:

(AΔB)ΔC=AΔ(BΔC).

Given information:

A,B and C be any sets.

Concept used:

AΔB=(A−B)∪(B−A)

Calculation:

The symmetric difference of the sets A and B is defined as,

AΔB=(A−B)∪(B−A)

Since

(A−B)∪(B−A)=(A∩BC)∪(B∩AC) by set difference={A∪(B∩ A C)}∩{BC∪(B∩ A C)} by the distributive law={(A∪B)∩(A∪ A C)}∩{( B C∪B)∩( B C∪ A C)}by the distributive law

={(A∪B)∩U}∩{U∩( B C∪ A C)}  by the complement laws=(A∪B)∩(B∩A)C by De Morgan's law=(A∪B)−(A∩B) by the set difference law

Therefore,

(A−B)∪(B−A)=(A∪B)−(A∩B)  .......(1)

Consider,

(AΔB)ΔC={(A−B)∪(B−A)}ΔC by the definition of Δ.={(A∪B)−(A∩B)}ΔC from equation (1)

={(A∪B)∩( A∩B)C}ΔC by the set difference law.={{( A∪B)∩ ( A∩B )C}−C}∪{C−{( A∪B)∩ ( A∩B )C}}by the definition of Δ .

={{( A∪B)∩ ( A∪B )C}∪C}−{{( A∪B)∩ ( A∪B )C}∩C} from equation (1)={{( A∪B)∩ ( A∪B )C}∪C}∩{{( A∪B)∩ ( A∪B ) C}∩C}Cby the set difference law