Question

Let Sbe a set and let P(S) be the set of all subsets of S. Define addition and multiplication in P(S) by the rules

$\mathit{M}\mathbf{+}\mathit{N}\mathbf{=}\mathbf{(}\mathbf{M}\mathbf{-}\mathbf{N}\mathbf{)}\mathbf{\cup }\mathbf{(}\mathbf{N}\mathbf{-}\mathbf{M}\mathbf{)}\mathbf{}\mathit{a}\mathit{n}\mathit{d}\mathbf{}\mathit{M}\mathit{N}\mathbf{=}\mathit{M}\mathbf{\cap }\mathit{N}$.

(b) Show that every element of P(S) satisfies the equations x² = x and x + x = 0_P(s) .

Short answer

It is proved that every element of P(S) satisfies the equationsx² = x and x + x = 0_P(s) .

Step-by-step solution

Prove that P(S) satisfies the equation x2 = x 

Consider M to be the element of P(S).

Prove that M² = M as:

$\begin{array}{rcl}{M}^2& =& M\cap M\\ & =& M\end{array}$

This implies that P(S) satisfies the equation x² = x.

Prove that P(S) satisfies the equation x + x = 0P(s) .

Solve for M + Mas:

$\begin{array}{rcl}M+M& =& \left(M-M\right)\cup \left(M-M\right)\\ & =& \varnothing \cup \varnothing \\ & =& \varnothing \end{array}$

This implies that P(S) satisfies the equation x + x = 0_P(s) .

Hence it is proved that every element of P(S) satisfies the equationsx² = x and x + x = 0_P(s) .