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