Laws of Boolean Algebra

According to the laws of Boolean Algebra, we use some variables, i.e. (A, B, C, X, Y, Z), and Apply NOT, AND, OR operations on these variables. 

Types of Laws in Boolean Algebra 

There are several laws of Boolean algebra, which are given below; let’s explain all equations of these laws with proof through the Truth Table.

1. Commutative Law

According to commutative law

• Any order in which two variables are AND’ed makes a similar result always (i.e., A . B = B . A )
• Any order in which two variables are OR’ed makes a similar result always (i.e., A + B = B + A)

2. Associative Law

• A.(B.C) = (A.B).C = A . B . C    (AND Associate Law)
• A + (B + C) = (A + B) + C = A + B + C    (OR Associate Law)

3. Distributive Law

• A.(B + C) = (A.B) + (A.C)    (OR Distributive Law)
• A + (B.C) = (A + B) . (A + C)    (AND Distributive Law)

4. Annulment law

According to Annulment Law

• A variable AND’ed with 0 will always be equal to 0 (i.e., A . 0 = 0)
• A variable OR’ed with 1 will always be equal to 1 (i.e.  A + 1 = 1)

5. Identity law

According to Identity Law

• A variable OR’ed with 0 will always be equal to that variable (i.e., A + 0 = A)
• A variable AND’ed with 1 will always be equal to that variable (i.e., A . 1 = A)

6. Idempotent law

According to idempotent law

• A variable OR’ed with itself will always be equal to that variable (i.e., A + A = A)
• A variable AND’ed with itself will always be equal to that variable (i.e., A . A = A )

7. Complement law

According to complement law

•   A variable AND’ed with its complement will always equal to 0. (i.e., A . A’ = 0)  
•  A variable OR’ed with its complement will always equal to 1. (i.e. A + A’ = 1)

8. Double Negation law

According to the double negation law

• The double complement of a variable will always equal that variable. (i.e. (A’)’ =A)

9. Absorption law

• A + (A.B) = A  (OR Absorption Law)
• A . (A+B) = A  (AND Absorption Law)

10. De Morgan’s Law

According to De Morgan’s law

• (A.B)’ = A’ + B’
• (A+B)’ = A’. B’