Laws of Algebra of logical statements - SS2 Mathematics Lesson Note

Let \(p,q,r\) be logical statements, then the following laws hold:

1. Commutative Laws:

   1. \(p \land q = q \land p\)

   2. \(p \vee q = q \vee p\)

2. Associative Laws:

   1. \((p \land q) \land r = p \land (q \land r)\)

   2. \((p \vee q) \vee r = p \vee (q \vee r)\)

3. Distributive Laws:

   1. \(p \land (q \vee r) = (p \land q) \vee (p \land r)\)

   2. \(p \vee (q \land r) = (p \vee q) \land (p \vee r)\)

4. Idempotent Laws:

   1. \(p \land p = p\)

   2. \(p \vee p = p\)

5. Identity Laws:

   1. \(p \land T = p\)

   2. \(p \land F = F\)

   3. \(p \vee T = T\)

   4. \(p \vee F = p\)

6. DeMorgan’s Laws:

   1. \(\sim(p \land q) = \sim p \vee \sim q\)

   2. \(\sim(p \vee q) = \sim p \land \sim q\)

7. Complement Laws:

   1. \(p \land \sim p = F\)

   2. \(p \vee \sim p = T\)

   3. \(\sim(\sim p) = p\)

   4. ~T=F; ~F=T

8. Law of Syllogism (Chain Rule):

   1. \((p \rightarrow q) \land (q \rightarrow r) \rightarrow (q \rightarrow r)\)