Answer to Question #103711 in Algebra for Ichigo

Question

For any two subsets A and B of a set U , we define their symmetric difference to
be A ∆ B = (A \ B) ∪ (B \ A)
i) Check whether ∆ distributes over ∩ .
ii) Show that A ∆(fi)=A.
iii) Prove that A∆B= (A∩ B' ) U (A' ∩B)

Answer 1

i) A∆B does distribute over A∩B because symmetric difference is equivalent to the union in both relative complements. The equality in this non-strict inclusion has occurred because there are disjoint sets. The symmetric difference is thus commutative and associative.

ii)

An empty set is a neutral set and

A∆ Ø = (A \ Ø) U (Ø \A)=A UØ=A

iii)

A∆ B = (A \ B) U (B \A)= (A ∩ B') U (B ∩ A')= (A ∩ B') U (A' ∩ B).

The symmetric difference in A∆B means that all elements belong to A or B but not in their intersection.