Sunday, October 2, 2011

EQ 5 Implication Axioms and Theorems

(1) Axiom: Definition of Implication: p $$\Rightarrow$$ q $$\hspace{0.2 cm }$$$$\equiv$$$$\hspace{0.2 cm }$$ p $$\vee$$ q $$\equiv$$ q

(2) Axiom: Consequence p $$\Leftarrow$$ q $$\hspace{0.2 cm }$$$$\equiv$$$$\hspace{0.2 cm }$$ q $$\Rightarrow$$ p

(3) Theorem: Definition of Implication p $$\Rightarrow$$ q $$\hspace{0.2 cm }$$$$\equiv$$$$\hspace{0.2 cm }$$ $$\neg$$p $$\vee$$ q

(4) Theorem: Definition of Implication: p $$\Rightarrow$$ q $$\hspace{0.2 cm }$$$$\equiv$$$$\hspace{0.2 cm }$$ p $$\wedge$$ q $$\equiv$$ p

(5) Theorem: Contrapositive p $$\Rightarrow$$ q $$\hspace{0.2 cm }$$$$\equiv$$$$\hspace{0.2 cm }$$ $$\neg$$q $$\Rightarrow$$ $$\neg$$ p

(6) Theorem: p $$\Rightarrow$$ (q $$\equiv$$ r)$$\hspace{0.2 cm }$$$$\equiv$$$$\hspace{0.2 cm }$$ p $$\wedge$$ q $$\equiv$$ p $$\wedge$$ r

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

(8) Theorem: Distributivity $$\frac{\Rightarrow}{\Rightarrow}$$ p $$\Rightarrow$$ (q $$\Rightarrow$$ r) $$\hspace{0.2 cm }$$$$\equiv$$$$\hspace{0.2 cm }$$ (p $$\Rightarrow$$ q) $$\Rightarrow$$ (p $$\Rightarrow$$ r)

(9) Theorem: Shunting p $$\wedge$$ q $$\Rightarrow$$ r $$\hspace{0.2 cm }$$$$\equiv$$$$\hspace{0.2 cm }$$ p $$\Rightarrow$$ (q $$\Rightarrow$$ r )

(10) Theorem: p $$\wedge$$ ( p $$\Rightarrow$$ q ) $$\hspace{0.2 cm }$$$$\equiv$$$$\hspace{0.2 cm }$$ p $$\wedge$$ q

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

(12) Theorem: p $$\vee$$ ( p $$\Rightarrow$$ q ) $$\hspace{0.2 cm }$$$$\equiv$$$$\hspace{0.2 cm }$$ true

(13) Theorem: p $$\vee$$ ( q $$\Rightarrow$$ p) $$\hspace{0.2 cm }$$$$\equiv$$$$\hspace{0.2 cm }$$ q $$\Rightarrow$$ p

(14) Theorem: p $$\vee$$ q $$\hspace{0.2 cm }$$$$\Rightarrow$$$$\hspace{0.2 cm }$$ p $$\wedge$$ q $$\equiv$$ p $$\equiv$$ q

(15) Reflexivity of $$\Rightarrow$$ : p $$\Rightarrow$$ p $$\hspace{0.2 cm }$$$$\equiv$$$$\hspace{0.2 cm }$$ true

(16) Theorem: Right Zero of $$\Rightarrow$$ : p $$\Rightarrow$$ true $$\hspace{0.2 cm }$$$$\equiv$$$$\hspace{0.2 cm }$$ true

(17) Theorem: Left Identity of $$\Rightarrow$$ : true $$\Rightarrow$$ p $$\hspace{0.2 cm }$$$$\equiv$$$$\hspace{0.2 cm }$$ p

(18) Theorem: p $$\Rightarrow$$ false $$\hspace{0.2 cm }$$$$\equiv$$$$\hspace{0.2 cm }$$ $$\neg$$p

(19) Theorem: false $$\Rightarrow$$ p $$\hspace{0.2 cm }$$$$\equiv$$$$\hspace{0.2 cm }$$ true

Weakening and Strengthening Theorems

(20) Theorem: p $$\hspace{0.2 cm }$$ $$\Rightarrow$$ $$\hspace{0.2 cm }$$ p $$\vee$$ q

(21) Theorem: p $$\wedge$$ q $$\hspace{0.2 cm }$$ $$\Rightarrow$$ $$\hspace{0.2 cm }$$ p

(22) Theorem: p $$\wedge$$ q $$\hspace{0.2 cm }$$ $$\Rightarrow$$ $$\hspace{0.2 cm }$$ p $$\vee$$ q

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

(24) Theorem: p $$\vee$$ (q $$\wedge$$ r) $$\hspace{0.2 cm }$$ $$\Rightarrow$$ $$\hspace{0.2 cm }$$ p $$\vee$$ q

(25) Theorem: Modus Ponens p $$\wedge$$ ( p $$\Rightarrow$$ q) $$\hspace{0.2 cm }$$ $$\Rightarrow$$ $$\hspace{0.2 cm }$$ q

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

(27) Theorem : (p $$\Rightarrow$$ r) $$\wedge$$ ( $$\neg$$p $$\Rightarrow$$ r) $$\hspace{0.2 cm }$$ $$\equiv$$ $$\hspace{0.2 cm }$$ r

(28) Theorem: Mutual Implication (p $$\Rightarrow$$ q) $$\wedge$$ (q $$\Rightarrow$$ p) $$\hspace{0.2 cm }$$ $$\equiv$$ $$\hspace{0.2 cm }$$ ( p $$\equiv$$ q)

(29) Theorem: Anti-Symmetry (p $$\Rightarrow$$ q) $$\wedge$$ (q $$\Rightarrow$$ p) $$\hspace{0.2 cm }$$ $$\Rightarrow$$ $$\hspace{0.2 cm }$$ ( p $$\equiv$$ q)

(30) Theorem: Transitivity (1) (p $$\Rightarrow$$ q) $$\wedge$$ (q $$\Rightarrow$$ r) $$\hspace{0.2 cm }$$ $$\Rightarrow$$ $$\hspace{0.2 cm }$$ (p $$\Rightarrow$$ r)

(31) Theorem: Transitivity (2) (p $$\equiv$$ q) $$\wedge$$ (q $$\Rightarrow$$ r) $$\hspace{0.2 cm }$$ $$\Rightarrow$$ $$\hspace{0.2 cm }$$ (p $$\Rightarrow$$ r)

(32) Theorem: Transitivity (3) (p $$\Rightarrow$$ q) $$\wedge$$ (q $$\equiv$$ r) $$\hspace{0.2 cm }$$ $$\Rightarrow$$ $$\hspace{0.2 cm }$$ (p $$\Rightarrow$$ r)

Other useful Implication Theorems

(33) Theorem: p $$\Rightarrow$$ ( q $$\Rightarrow$$ p) $$\hspace{0.2 cm }$$ $$\Rightarrow$$ $$\hspace{0.2 cm }$$ true

(34) Theorem (p $$\Rightarrow$$ q) $$\hspace{0.2 cm }$$ $$\Rightarrow$$ $$\hspace{0.2 cm }$$ (p $$\wedge$$ r) $$\Rightarrow$$ (q $$\wedge$$ r)

(35) Theorem (p $$\Rightarrow$$ q) $$\wedge$$ (r $$\Rightarrow$$ s) $$\hspace{0.2 cm }$$ $$\Rightarrow$$ $$\hspace{0.2 cm }$$ (p $$\vee$$ r) $$\Rightarrow$$ (q $$\vee$$ s)

(36) Theorem (p $$\Rightarrow$$ q) $$\wedge$$ (r $$\Rightarrow$$ s) $$\hspace{0.2 cm }$$ $$\Rightarrow$$ $$\hspace{0.2 cm }$$ (p $$\wedge$$ r) $$\Rightarrow$$ (q $$\wedge$$ s)

(37) Theorem a $$\Rightarrow$$ $$\neg$$b $$\hspace{0.2 cm }$$ $$\equiv$$ $$\hspace{0.2 cm }$$ a $$\Rightarrow$$ b $$\equiv$$ $$\neg$$a

(38) Theorem $$\neg$$a $$\Rightarrow$$ b $$\hspace{0.2 cm }$$ $$\equiv$$ $$\hspace{0.2 cm }$$ a $$\Rightarrow$$ b $$\equiv$$ b