EQ 12 Conjunction

(1) Axiom: Golden Rule: p $\wedge$ q $\hspace{0.2 cm} \equiv \hspace{0.2 cm}$ p $\vee$ q $\equiv$ p $\equiv$ q

(2) Theorem Symmetry of $\wedge$: p $\wedge$ q $\hspace{0.2 cm} \equiv \hspace{0.2 cm}$ q $\wedge$ p

(3) Theorem Associativity of $\wedge$: p $\wedge$ ( q $\wedge$ r) $\hspace{0.2 cm} \equiv \hspace{0.2 cm}$ (p $\wedge$ q) $\wedge$ r

(4) Theorem Idempotency of $\wedge$: p $\wedge$ p $\hspace{0.2 cm} \equiv \hspace{0.2 cm}$ p


(5) Theorem Identity of $\wedge$: p $\wedge$ true $\hspace{0.2 cm} \equiv \hspace{0.2 cm}$ p


(6) Theorem Zero of $\wedge$: p $\wedge$ false $\hspace{0.2 cm} \equiv \hspace{0.2 cm}$ false


(7) Theorem Distributivity of $\frac{\wedge}{\wedge}$: p $\wedge$ ( q $\wedge$ r) $\hspace{0.2 cm} \equiv \hspace{0.2 cm}$ (p $\wedge$ q) $\wedge$ (p $\wedge$ r)

(8) Theorem Contradiction: p $\wedge$ $\neg$p $\hspace{0.2 cm} \equiv \hspace{0.2 cm}$ false

(9) Theorem Absorption 1: p $\wedge$ ( p $\vee$ q) $\hspace{0.2 cm} \equiv \hspace{0.2 cm}$ p


(10) Theorem Absorption 2: p $\wedge$ ($\neg$ p $\vee$ q) $\hspace{0.2 cm} \equiv \hspace{0.2 cm}$ p $\wedge$ q

(11) Theorem Absorption 3: p $\vee$ ( p $\wedge$ q) $\hspace{0.2 cm} \equiv \hspace{0.2 cm}$ p

(12) Theorem Absorption 4: p $\vee$ ($\neg$ p $\wedge$ q) $\hspace{0.2 cm} \equiv \hspace{0.2 cm}$ p $\vee$ q

(13) Theorem Distributivity of $\frac{\wedge}{\vee}$: p $\wedge$ ( q $\vee$ r) $\hspace{0.2 cm} \equiv \hspace{0.2 cm}$ (p $\wedge$ q) $\vee$ (p $\wedge$ r)

(14) Theorem Distributivity of $\frac{\vee}{\wedge}$: p $\vee$ ( q $\wedge$ r) $\hspace{0.2 cm} \equiv \hspace{0.2 cm}$ (p $\vee$ q) $\wedge$ (p $\vee$ r)

(15) Theorem De Morgan 1 $\neg$ (p $\wedge$ q) $\hspace{0.2 cm} \equiv \hspace{0.2 cm}$ ($\neg$p $\vee$$\neg$ q)

(16) Theorem De Morgan 2 $\neg$ (p $\vee$ q) $\hspace{0.2 cm} \equiv \hspace{0.2 cm}$ ($\neg$p $\wedge$$\neg$ q)

(17) Theorem p $\wedge$ q $\hspace{0.2 cm} \equiv \hspace{0.2 cm}$ p $\wedge$ $\neg$q $\equiv$$\neg$ p

(18) Theorem p $\wedge$ ( q $\equiv$ r) $\hspace{0.2 cm} \equiv \hspace{0.2 cm}$ p $\wedge$ q $\equiv$ p $\wedge$ r $\equiv$ p

(19) Theorem p $\wedge$ ( q $\equiv$ p) $\hspace{0.2 cm} \equiv \hspace{0.2 cm}$ p $\wedge$ q

(20) Theorem Replacement ( p $\equiv$ q) $\wedge$ ( r $\equiv$ p) $\hspace{0.2 cm} \equiv \hspace{0.2 cm}$ ( p $\equiv$ q) $\wedge$ ( r $\equiv$ q)

(21) Theorem Definition of $\equiv$: ( p $\equiv$ q) $\hspace{0.2 cm} \equiv \hspace{0.2 cm}$ ( p $\wedge$ q) $\vee$ ($\neg$ p $\wedge$ $\neg$q)

(22) Theorem Exclusive Or: ( p $\not\equiv$ q) $\hspace{0.2 cm} \equiv \hspace{0.2 cm}$ ( p $\wedge$ $\neg$q) $\vee$ ($\neg$ p $\wedge$ q)


(23) Theorem (p $\wedge$ q) $\wedge$ r $\hspace{0.2 cm} \equiv \hspace{0.2 cm}$ p $\equiv$ q $\equiv$ r $\equiv$ p $\vee$ q $\equiv$ q $\vee$ r $\equiv$ p $\vee$ r $\equiv$ p $\vee$ q $\vee$ r