Laws Of Algebra Of Propositions

Laws of algebra of propositions

1. Idempotent laws

a) $P \lor P \equiv P$

b) $P \land P \equiv P$

2. Commutative

a) $P \lor Q \equiv Q \lor P$

b) $P \land Q \equiv Q \land P$

3. Associative laws

a) $(P \lor Q) \lor R \equiv P \lor (Q \lor R)$

b) $(P \land Q) \land R \equiv P \land (Q \land R)$

4. Distributive laws

a) $P \lor (Q \land R) \equiv (P \lor Q) \land (P \lor R)$

b) $P \land (Q \lor R) \equiv (P \land Q) \lor (P \land R)$

5. Identity laws

a) $P \lor F \equiv P$

b) $P \land T \equiv P$

c) $P \lor T \equiv T$

d) $P \land F \equiv F$

6. Complementary laws

a) $P \lor \neg P \equiv T$

b) $P \land \neg P \equiv F$

c) $\neg \neg P \equiv P$

d) $\neg T \equiv F$

e) $\neg F \equiv T$

7. De-Morgan's law

a) $\neg (P \lor Q) \equiv \neg P \land \neg Q$

b) $\neg (P \land Q) \equiv \neg P \lor \neg Q$

Examples

Using the laws of algebra of proposition simplify $(P \lor Q) \land \neg P$

Solution

$(P \lor Q) \land \neg P \equiv (\neg P \land P) \lor (\neg P \land Q)$ ……distributive law

$\equiv F \lor (\neg P \land Q)$ ………complement law

$\equiv (\neg P \land Q)$ ………..identity

Questions

1. Simplify the following propositions using the laws of algebra of propositions

i) $\neg (P \lor Q) \lor (\neg P \land Q)$

ii) $(P \land Q) \lor [\neg R \land (Q \land P)]$

2. Show using the laws of algebra of propositions $(P \land Q) \lor \equiv P$

3. Construct a truth table for $[(p \to \neg q) \land (r \to p) \land r] \to \neg p$