## Wednesday, October 5, 2011

### 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