Question

On the power set P of a non-empty set A, we define an operation ∆ by $X∆Y=\left(\overline{X}\cap Y\right)\cup \left(X\cap \overline{Y}\right)$
Then which are of the following statements is true about ∆
(a) commutative and associative without an identity
(b) commutative but not associative with an identity
(c) associative but not commutative without an identity
(d) associative and commutative with an identity

Answer 1

(d) associative and commutative with an identity

$$\begin{gathered} \mathrm{Commutativity}: \\ X∆Y=\left(\overline{X}\cap Y\right)\cup \left(X\cap \overline{Y}\right) \\ =\left(\overline{Y}\cap X\right)\cup \left(Y\cap \overline{X}\right) \\ =Y∆X \\ \mathrm{Thus}, \\ X∆Y=Y∆X \\ \mathrm{Hence},∆\mathrm{is}\mathrm{commutative}\mathrm{on}A. \end{gathered}$$

Let $\varphi$ be the identity element for $∆$ on P.
$$\begin{gathered} A∆\varphi =\left(\overline{A}\cap \varphi \right)\cup \left(A\cap \overline{\varphi }\right) \\ =\varphi \cup A \\ =A \\ \mathrm{and}, \\ \varphi ∆A=\left(\overline{\varphi }\cap A\right)\cup \left(\varphi \cap \overline{A}\right) \\ =A\cup \varphi \\ =A \end{gathered}$$