Definitions

\(\displaystyle{p}\leftrightarrow{q}\equiv{\left({p}\wedge{q}\right)}\vee{\left(\neg{p}\wedge\neg{q}\right)}\) (1)

\(\displaystyle{p}\oplus\equiv{\left({p}\vee{q}\right)}\vee\neg{\left({p}\wedge{q}\right)}\) (2)

Double negation law: \(\displaystyle\neg{\left(\neg{p}\right)}\equiv{p}\)

Commutative laws: \(\displaystyle{p}\vee{q}\equiv{q}\vee{p},{p}\wedge{q}\equiv{q}\wedge{p}\)

De Morgan's laws: \(\displaystyle\neg{\left({p}\wedge{q}\right)}\equiv\neg{p}\vee\neg{q},\neg{\left({p}\vee{q}\right)}\equiv\neg{p}\wedge\neg{q}\)

a) \(\displaystyle{\left(\neg{p}\leftrightarrow\neg{q}\right)}\)

\(\displaystyle\equiv{\left(\neg{p}\wedge\neg{q}\right)}\vee{\left(\neg{\left(\neg{p}\right)}\wedge{\left(\neg{\left(\neg{q}\right)}\right)}\right.}\)

\(\displaystyle\equiv{\left(\neg{p}\wedge\neg{q}\right)}\vee{\left({p}\vee{q}\right)}\)

\(\displaystyle\equiv{\left({p}\wedge{q}\right)}\vee{\left(\neg{p}\wedge\neg{q}\right)}\)

\(\displaystyle\equiv{p}\leftrightarrow{q}\)

b) \(\displaystyle\neg{\left({p}\oplus{q}\right)}\)

\(\displaystyle\equiv\neg{\left({\left({p}\vee{q}\right)}\wedge\neg{\left({p}\wedge{q}\right)}\right.}\)

\(\displaystyle\equiv{\left(\neg{\left({p}\vee{q}\right)}\right)}\vee{\left(\neg{\left(\neg{\left({p}\wedge{q}\right)}\right)}\right)}\)

\(\displaystyle\equiv{\left(\neg{\left({p}\vee{q}\right)}\right)}{v}{\left({p}\wedge{q}\right)}\)

\(\displaystyle\equiv{\left({\left(\neg{p}\right)}\wedge{\left(\neg{q}\right)}\right)}\vee{\left({p}\wedge{q}\right)}\)

\(\displaystyle\equiv{\left({p}\wedge{q}\right)}\vee{\left({\left(\neg{p}\wedge{\left(\neg{q}\right)}\right)}\right.}\)

\(\displaystyle\equiv{p}\leftrightarrow{q}\)

\(\displaystyle{p}\leftrightarrow{q}\equiv{\left({p}\wedge{q}\right)}\vee{\left(\neg{p}\wedge\neg{q}\right)}\) (1)

\(\displaystyle{p}\oplus\equiv{\left({p}\vee{q}\right)}\vee\neg{\left({p}\wedge{q}\right)}\) (2)

Double negation law: \(\displaystyle\neg{\left(\neg{p}\right)}\equiv{p}\)

Commutative laws: \(\displaystyle{p}\vee{q}\equiv{q}\vee{p},{p}\wedge{q}\equiv{q}\wedge{p}\)

De Morgan's laws: \(\displaystyle\neg{\left({p}\wedge{q}\right)}\equiv\neg{p}\vee\neg{q},\neg{\left({p}\vee{q}\right)}\equiv\neg{p}\wedge\neg{q}\)

a) \(\displaystyle{\left(\neg{p}\leftrightarrow\neg{q}\right)}\)

\(\displaystyle\equiv{\left(\neg{p}\wedge\neg{q}\right)}\vee{\left(\neg{\left(\neg{p}\right)}\wedge{\left(\neg{\left(\neg{q}\right)}\right)}\right.}\)

\(\displaystyle\equiv{\left(\neg{p}\wedge\neg{q}\right)}\vee{\left({p}\vee{q}\right)}\)

\(\displaystyle\equiv{\left({p}\wedge{q}\right)}\vee{\left(\neg{p}\wedge\neg{q}\right)}\)

\(\displaystyle\equiv{p}\leftrightarrow{q}\)

b) \(\displaystyle\neg{\left({p}\oplus{q}\right)}\)

\(\displaystyle\equiv\neg{\left({\left({p}\vee{q}\right)}\wedge\neg{\left({p}\wedge{q}\right)}\right.}\)

\(\displaystyle\equiv{\left(\neg{\left({p}\vee{q}\right)}\right)}\vee{\left(\neg{\left(\neg{\left({p}\wedge{q}\right)}\right)}\right)}\)

\(\displaystyle\equiv{\left(\neg{\left({p}\vee{q}\right)}\right)}{v}{\left({p}\wedge{q}\right)}\)

\(\displaystyle\equiv{\left({\left(\neg{p}\right)}\wedge{\left(\neg{q}\right)}\right)}\vee{\left({p}\wedge{q}\right)}\)

\(\displaystyle\equiv{\left({p}\wedge{q}\right)}\vee{\left({\left(\neg{p}\wedge{\left(\neg{q}\right)}\right)}\right.}\)

\(\displaystyle\equiv{p}\leftrightarrow{q}\)