## Wednesday, October 5, 2011

### EQ 14 Negation, Inequivalence, False

(1) Axiom: Definition of false: false $$\hspace{0.2 cm} \equiv \hspace{0.2 cm}$$ $$\neg$$ true

(2) Axiom: Distributivity of $$\frac{\neg}{\equiv}$$: $$\neg$$ (p $$\equiv$$ q) $$\hspace{0.2 cm} \equiv \hspace{0.2 cm}$$ $$\neg$$p $$\equiv$$ q

(3) Axiom: Definition $$\not\equiv$$: (p $$\not\equiv$$ q) $$\hspace{0.2 cm} \equiv \hspace{0.2 cm}$$ $$\neg$$(p $$\equiv$$ q)

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

(5) Theorem: Double Negation: $$\neg$$$$\neg$$p $$\hspace{0.2 cm} \equiv \hspace{0.2 cm}$$ p

(6) Theorem: Negation of false: $$\neg$$false $$\hspace{0.2 cm} \equiv \hspace{0.2 cm}$$ true

(7) Theorem: (p $$\not\equiv$$ q) $$\hspace{0.2 cm} \equiv \hspace{0.2 cm}$$ $$\neg$$p $$\equiv$$ q

(8) Theorem: $$\neg$$p$$\equiv$$p$$\equiv$$false

(9) Theorem: Symmetry of $$\not\equiv$$: p $$\not\equiv$$ q $$\hspace{0.2 cm} \equiv \hspace{0.2 cm}$$ q $$\not\equiv$$ p

(10) Theorem: Associativity of $$\not\equiv$$: (p $$\not\equiv$$ (q $$\not\equiv$$ r)) $$\hspace{0.2 cm} \equiv \hspace{0.2 cm}$$ ((p $$\not\equiv$$ q ) $$\not\equiv$$ r)

(11) Mutual Associativity: (p $$\not\equiv$$ (q $$\equiv$$ r)) $$\hspace{0.2 cm} \equiv \hspace{0.2 cm}$$ ((p $$\not\equiv$$ q ) $$\equiv$$ r)

(12) Mutual Interchangability: p $$\not\equiv$$ q $$\equiv$$ r $$\hspace{0.2 cm} \equiv \hspace{0.2 cm}$$ p $$\equiv$$ q $$\not\equiv$$ r