Commutative:

For \(\displaystyle{x}\in{P}{\left({A}\right)}{\quad\text{and}\quad}{y}\in{P}{\left({A}\right)},\)

\(\displaystyle{x}\oplus{y}={\left({x}\cap{y}^{{c}}\right)}\cup{\left({y}\cap{x}^{{c}}\right)}{\quad\text{and}\quad}\)

\(\displaystyle{y}\oplus{x}={\left({y}\cap{x}^{{c}}\right)}\cup{\left({x}\cap{y}^{{c}}\right)}\)

\(\displaystyle={\left({x}\cap{y}^{{c}}\right)}\cup{\left({y}\cap{x}^{{c}}\right)}\)

\(\displaystyle={x}\oplus{y}\)

Associative

For \(\displaystyle{x},{y}{\quad\text{and}\quad}{z}\in{P}{\left({A}\right)}\)

\(\displaystyle{\left({x}\oplus{y}\right)}\oplus{z}={\left({\left({x}\oplus{y}\right)}\cap{z}^{{c}}\right)}\cup{\left({z}{n}{\left({x}\oplus{y}\right)}^{{c}}\right)}\)

\(\displaystyle={\left({z}\cap{\left({\left({x}\cap{y}^{{c}}\right)}\cup{\left({y}\cap{x}^{{c}}\right)}^{{c}}\right)}\cup{\left({\left({\left({x}\cap{y}^{{c}}\right)}\cup{\left({y}\cap{x}^{{c}}\right)}\right)}\cap{z}^{{c}}\right)}\right.}\)

\(\displaystyle={\left({z}\cap{\left({x}^{{c}}\cup{y}\right)}\cap{\left({y}^{{c}}\cup{x}\right)}\right)}\cup{\left({\left({x}\cap{y}^{{c}}\right)}\cap{z}^{{c}}\right)}\cup{\left({\left({y}\cap{x}^{{c}}\right)}\cap{z}^{{c}}\right)}\)

\(\displaystyle={\left({z}\cap{\left({x}^{{c}}\cap{\left({y}^{{c}}\cup{x}\right)}\right)}\cup{\left({\left({y}\cap{y}^{{c}}\right)}\cup{x}\right)}\right)}\cup{\left({\left({\left({x}\cap{y}^{{c}}\right)}\cup{\left({y}\cap{x}^{{c}}\right)}\right)}\cap{z}^{{c}}\right)}\)

\(\displaystyle={\left({z}\cap{\left({\left({x}^{{c}}\cap{y}^{{c}}\right)}\cup{\left({x}^{{c}}\cap{x}\right)}\right)}\cup{\left({\left({y}\cap{y}^{{c}}\right)}\cup{\left({y}\cap{x}\right)}\right)}\right)}\cup{\left({\left({\left({x}\cap{y}^{{}}\right)}\cap{z}^{{c}}\right)}\cup{\left({\left({y}\cap{x}^{{c}}\right)}\cap{z}^{{c}}\right)}\right)}\)

\(\displaystyle={\left({z}\cap{\left({\left({x}^{{c}}\cap{y}^{{c}}\right)}\cup{\left({x}^{{c}}\cap{x}\right)}\right)}{u}{\left({\left({y}\cap{y}^{{c}}\right)}\cup{\left({y}\cap{x}\right)}\right)}\right)}\cup{\left({x}\cap{y}^{{c}}\cap{z}^{{c}}\right)}\cup{\left({y}\cap{x}^{{c}}\cap{z}^{{c}}\right)}\)

\(\displaystyle={\left({z}\cap{x}^{{c}}\cap{y}^{{c}}\right)}\cup{\left({z}\cap{y}\cap{x}\right)}\cup{\left({x}\cap{y}^{{c}}\cap{z}^{{c}}\right)}\cup{\left({y}\cap{x}^{{c}}\cap{z}^{{c}}\right)}\)

Similarly, it can be proved that

\(\displaystyle{x}\oplus{\left({y}\oplus{z}\right)}={\left({z}\cap{x}^{{c}}\cap{y}^{{c}}\right)}\cup{\left({z}\cap{y}\cap{x}\right)}\cup{\left({x}\cap{y}^{{c}}\cap{z}^{{c}}\right)}\cup{\left({y}\cap{x}^{{c}}\cap{z}^{{c}}\right)}\)

For \(\displaystyle{x}\in{P}{\left({A}\right)}{\quad\text{and}\quad}{y}\in{P}{\left({A}\right)},\)

\(\displaystyle{x}\oplus{y}={\left({x}\cap{y}^{{c}}\right)}\cup{\left({y}\cap{x}^{{c}}\right)}{\quad\text{and}\quad}\)

\(\displaystyle{y}\oplus{x}={\left({y}\cap{x}^{{c}}\right)}\cup{\left({x}\cap{y}^{{c}}\right)}\)

\(\displaystyle={\left({x}\cap{y}^{{c}}\right)}\cup{\left({y}\cap{x}^{{c}}\right)}\)

\(\displaystyle={x}\oplus{y}\)

Associative

For \(\displaystyle{x},{y}{\quad\text{and}\quad}{z}\in{P}{\left({A}\right)}\)

\(\displaystyle{\left({x}\oplus{y}\right)}\oplus{z}={\left({\left({x}\oplus{y}\right)}\cap{z}^{{c}}\right)}\cup{\left({z}{n}{\left({x}\oplus{y}\right)}^{{c}}\right)}\)

\(\displaystyle={\left({z}\cap{\left({\left({x}\cap{y}^{{c}}\right)}\cup{\left({y}\cap{x}^{{c}}\right)}^{{c}}\right)}\cup{\left({\left({\left({x}\cap{y}^{{c}}\right)}\cup{\left({y}\cap{x}^{{c}}\right)}\right)}\cap{z}^{{c}}\right)}\right.}\)

\(\displaystyle={\left({z}\cap{\left({x}^{{c}}\cup{y}\right)}\cap{\left({y}^{{c}}\cup{x}\right)}\right)}\cup{\left({\left({x}\cap{y}^{{c}}\right)}\cap{z}^{{c}}\right)}\cup{\left({\left({y}\cap{x}^{{c}}\right)}\cap{z}^{{c}}\right)}\)

\(\displaystyle={\left({z}\cap{\left({x}^{{c}}\cap{\left({y}^{{c}}\cup{x}\right)}\right)}\cup{\left({\left({y}\cap{y}^{{c}}\right)}\cup{x}\right)}\right)}\cup{\left({\left({\left({x}\cap{y}^{{c}}\right)}\cup{\left({y}\cap{x}^{{c}}\right)}\right)}\cap{z}^{{c}}\right)}\)

\(\displaystyle={\left({z}\cap{\left({\left({x}^{{c}}\cap{y}^{{c}}\right)}\cup{\left({x}^{{c}}\cap{x}\right)}\right)}\cup{\left({\left({y}\cap{y}^{{c}}\right)}\cup{\left({y}\cap{x}\right)}\right)}\right)}\cup{\left({\left({\left({x}\cap{y}^{{}}\right)}\cap{z}^{{c}}\right)}\cup{\left({\left({y}\cap{x}^{{c}}\right)}\cap{z}^{{c}}\right)}\right)}\)

\(\displaystyle={\left({z}\cap{\left({\left({x}^{{c}}\cap{y}^{{c}}\right)}\cup{\left({x}^{{c}}\cap{x}\right)}\right)}{u}{\left({\left({y}\cap{y}^{{c}}\right)}\cup{\left({y}\cap{x}\right)}\right)}\right)}\cup{\left({x}\cap{y}^{{c}}\cap{z}^{{c}}\right)}\cup{\left({y}\cap{x}^{{c}}\cap{z}^{{c}}\right)}\)

\(\displaystyle={\left({z}\cap{x}^{{c}}\cap{y}^{{c}}\right)}\cup{\left({z}\cap{y}\cap{x}\right)}\cup{\left({x}\cap{y}^{{c}}\cap{z}^{{c}}\right)}\cup{\left({y}\cap{x}^{{c}}\cap{z}^{{c}}\right)}\)

Similarly, it can be proved that

\(\displaystyle{x}\oplus{\left({y}\oplus{z}\right)}={\left({z}\cap{x}^{{c}}\cap{y}^{{c}}\right)}\cup{\left({z}\cap{y}\cap{x}\right)}\cup{\left({x}\cap{y}^{{c}}\cap{z}^{{c}}\right)}\cup{\left({y}\cap{x}^{{c}}\cap{z}^{{c}}\right)}\)