## Saturday, November 5, 2011

### $$\equiv$$ Theorem 2: p $$\equiv$$ p

$$\hspace{2 cm}$$ p $$\equiv$$ p
$$\hspace{0.5 cm} =$$ the $$\equiv$$ Identity Axiom. $$p \equiv p \equiv true$$
$$\hspace{2 cm}$$ true. QED

### $$\equiv$$ Theorem 1: true

$$\hspace{2 cm} true \equiv p \equiv p$$, Axiom, Identity of $$\equiv$$,
$$\hspace{0.5 cm} =$$ Leibniz, E = Identity Axiom above, $$p = p \equiv p, q = true$$
$$\hspace{2 cm} true \equiv true$$
$$\hspace{0.5 cm} \equiv$$ Axiom, Identity of $$\equiv$$,$$true \equiv p \equiv p$$,
$$\hspace{2 cm}$$ true

(1) Axiom: Associativity of $$\equiv$$: p $$\equiv$$ (q $$\equiv$$ r) $$\hspace{0.2 cm} \equiv \hspace{0.2 cm}$$ (p $$\equiv$$ q ) $$\equiv$$ r
(2) Axiom: Symmetry of $$\equiv$$: p $$\equiv$$ q $$\hspace{0.2 cm} \equiv \hspace{0.2 cm}$$ q $$\equiv$$ p
(3) Axiom: Identity of $$\equiv$$: true $$\equiv$$ q $$\equiv$$ q