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