Abstract Algebra: Theory and Applications · Problem 1.20

Preliminaries — Thomas Judson

Mathematics — Abstract Algebra · set-theory-operations

Problem

Let

A

and

B

be sets. Define the symmetric difference of

A

and

B

to be

A \triangle B = (A \setminus B) \cup (B \setminus A)

. Prove the following statements. ((a))

A \triangle B = (A \cup B) \setminus (A \cap B)

; ((b))

A \triangle B = B \triangle A

; ((c))

(A \triangle B) \triangle C = A \triangle (B \triangle C)

; ((d))

A \triangle \emptyset = A

; ((e))

A \triangle A = \emptyset