Boolean Algebra

Boolean Algebra Laws

1. Identity Laws

• A + 0 = A
• A ⋅ 1= A

2. Null Laws

• A + 1 = 1
• A ⋅ 0 = 0

3. Idempotent Laws

• A + A = A
• A ⋅ A = A

4. Complement Laws

• A + A′ = 1
• A ⋅ A′ = 0

5. Domination Laws

• A+1 = 1
• A ⋅ 0 = 0

6. Double Negation

• (A′)′ = A

7. Commutative Laws

• A + B = B + A
• A ⋅ B = B ⋅ A

8. Associative Laws ⭐

• ( A + B ) + C = A + (B + C)
• ( A ⋅ B ) ⋅ C = `A . (B . C)

9. Distributive Laws ⭐

• A ( B + C ) = AB + AC
• A + BC = (A + B)(A + C)

12. Special Distributive Form ⭐

• X + X′Y = X + YX + X' Y = X + Y
• Example: B′+BA = B′+AB' + BA = B' + A

10. Absorption Laws ⭐

• A + AB = A
• A ( A + B ) = A

11. De Morgan’s Laws ⭐

• ( A + B )′ = A'B'
• (AB)′ = A' + B'

Proofs of Some Important Algebra

1. Distributive Laws

• (a) $A(B+C) = AB + AC$
• (b) $(A+B)(A+C) = A + BC$

$$\begin{aligned} (A + B)(A + C) &= A(A + C) + B(A + C) \ &= A + AC + AB + BC \ &= A + AB + AC + BC \ &= A (1 + B + C) + BC \ &= A + BC \end{aligned}$$

2. Special Distributive Laws

• (a) $X+X′Y= X+Y$

$$\begin{aligned} X + X’Y &= (X + X’)(X + Y) \ &= (1)(X + Y) \ &= X + Y \end{aligned}$$

3. Absorption Laws

• (a) $A+AB = A(1+B) = A(1) =A$
• (b) $A(A+B)=AA+AB=A+AB=A$

4. De Morgan’s Laws

• (a) $(A+B)′=A′B′$
• (b) $(AB)′=A′+B′$

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