CISC 102 - Day 5

2024-09-18 21:27:29 -0400 EDT

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More Logic

Definition: we say two logical statements are logically equivalent when identical inputs give identical outputs.

Example: Show $\neg (p \vee q) \equiv \neg p \wedge \neg q$

p	q	$\neg (p \vee q)$	$\neg p \wedge \neg q$
T	T	F	F
T	F	T	T
F	T	T	T
F	F	T	T

$\therefore \neg (p \vee q) \equiv \neg p \wedge \neg q$ because they give the same output form the same input.

Algebra for Logic

1. $ (p \wedge q) \wedge r \equiv p \wedge (q \wedge r) $
2. $ p \vee q \equiv q \vee p $
3. $ p \wedge (q \vee r) \equiv (p \wedge q ) \vee (p \wedge r) $
4. $ \neg (p) \equiv \neg p \wedge \neg q $
5. $ \neg (\neg p) \equiv p $
6. $ \neg (p \wedge p) \equiv $ False
7. $ \neg (p \vee p) \equiv $ True

Also: $ p \rightarrow q \equiv q \vee \neg p $