Question

Show that if A and B are sets, then

a) \(A \oplus B{\bf{ = }}B \oplus A\)

b) \(\left( {A \oplus B} \right) \oplus B = A\)

Short answer

(a) \(A \oplus B = B \oplus A\)

(b) \((A \oplus B) \oplus B = A\)

Step-by-step solution

Solve \(A \oplus B\) using the identity\(A \oplus B = \left( {A - B} \right) \cup \left( {B - A} \right)\)

(a)

\(\begin{aligned}{c}A \oplus B = (A - B) \cup (B - A)\\ = (B - A) \cup (A - B)\\ = B \oplus A\end{aligned}\)

Solve \(\left( {A \oplus B} \right) \oplus B\) using the identity\(A \oplus B = \left( {A - B} \right) \cup \left( {B - A} \right)\)

(b)

\(\begin{aligned}{c}(A \oplus B) \oplus B = ((A - B) \cup (B - A) - B) \cup (B - (A - B) \cup (B - A))\\ = (A \cup A) \cup (A \cup A)\\ = A\end{aligned}\)

Hence, proved.