Logic Laws

Commutative

• $p\wedge q\equiv q\wedge p$
• $p\vee q\equiv q\vee p$

Associative

• $p\wedge q\wedge r\equiv (p\wedge q)\wedge r\equiv p\wedge (q\wedge r)$
• $p\vee q\vee r\equiv (p\vee q)\vee r\equiv p\vee (q\vee r)$

Distributive

• $p\wedge (q\vee r)\equiv (p\wedge q)\vee (p\wedge r)$
• $p\vee (q\wedge r)\equiv (p\vee q)\wedge (p\vee r)$
• applying twice: $(p\wedge q)\vee (r\wedge s)\equiv (p\vee r)\wedge (p\vee s)\wedge (q\vee r)\wedge (q\vee s)$ and vice versa

Identities

• $p\wedge true\equiv p$
• $p\vee false\equiv p$

Universal Bound

• $p\vee true\equiv true$
• $p\wedge false\equiv false$

Negation

• $p\vee \sim p\equiv true$
• $p\wedge \sim p\equiv false$

Double Negative

• $\sim (\sim p)\equiv p$

Idempotent

• $p\wedge p\equiv p$
• $p\vee p\equiv p$

Absorption

• $p\vee (p\wedge q)\equiv p$
• $p\wedge (p\vee q)\equiv p$

Implication

• $p\to q\equiv \sim p\vee q$
• $\sim (p\to q)\equiv p\wedge \sim q$ (Can be deduced from the point above)
• no name for this: $p\leftrightarrow q\equiv (\sim p\wedge \sim q)\vee (p\wedge q)$ this is derived from implication law, distributive law, negation law and identity law

De Morgan’s

These laws can be extended to more than two variables

• $\sim (p\wedge q)\equiv \sim p\vee \sim q$
• $\sim (p\vee q)\equiv \sim p\wedge \sim q$