Question

Definition 1. The symmetric difference of two sets A and B is the set A△B defined by A△B = (A \ B) ∪ (B \ A).
(a) Draw the Venn diagram for the symmetric difference.
(b) Prove that A△B = (A ∪ B) \ (A ∩ B).
(c) Prove A△A = ∅, A△∅ = A.
(d) Prove that for sets A, B, we have A△B = A \ B if and only if B ⊆ A.

Answer 1

c) Since there is no any element in AA i.e. AA is empty set.

A∆A is empty set.

Since every element of A is element of A(empty set)

and there is no element in (empty set)A

A∆(empty set) = A.

d) Since A∆B = (AB) U (BA)

Therefore A∆B = AB

iff BA is empty set (any x in BA cannot be in AB)

iff B is subset of A (because if B is not subset of A, then there must be x in B which is not in A)

Hence A∆B = AB iff B is subset of A.