Note: Every law in Boolean algebra has two forms that are obtained by exchanging all the ANDs to ORs and 1s to 0s and vice versa. This is known as the Boolean algebra duality principle.  The order of operations for Boolean algebra, from highest to lowest priority is NOT, then AND, then OR. Expressions inside brackets are always evaluated first.

1. Commutative Law

(a) A + B = B + A

(b) A $\cdot$ B = B $\cdot$ A

2. Associative Law

(a) (A + B) + C = A + (B + C)

(b) (A $\cdot$ B) $\cdot$ C = A $\cdot$ (B $\cdot$ C)

3. Distributive Law

(a) A $\cdot$ (B + C) = A $\cdot$ B + A $\cdot$ C

(b) A + (B $\cdot$ C) = (A + B)$\cdot$ (A + C)

4. Null Law

(a) 1 + A = 1   

(b) 0 $\cdot$ A = 0

5. Identity law

(a) 0 + A = A

(b) 1 $\cdot$ A = A

6. Idempotent Law

(a) A + A = A

(b) A $\cdot$ A = A

7. Complementarity Law

(a) A' + A = 1

(b) A' $\cdot$ A = 0

8. Uniting Theorem

(a) A $\cdot$ B + A $\cdot$ B' = A

(b) (A + B) $\cdot$ (A + B') = A

9. Absorption Theorem

(a) A + A $\cdot$ B = A

(b) A $\cdot$ (A + B) = A

10. Adsorption Theorem

(a)  A + A'$\cdot$ B = A + B

(b) A $\cdot$ (A' + B) = A $\cdot$ B

11. De Morgan's Theorem

(a) (A + B)' = A' $\cdot$ B'

(b) (A $\cdot$ B)' = A' + B'