# Boolean Algebra

In this blog post we are investigating different formulas than can be used to simplify a Boolean expression.

¬ ¬A = A

A ∧ ¬A = 0
A ∨ ¬A = 1

A ∧ A = A
A ∨ A = A

A ∧ 1 = A
A ∧ 0 = 0
A ∨ 1 = 1
A ∨ 0 = A

#### Associative Laws

(A ∧ B) ∧ C = A ∧ (B ∧ C)
(A ∨ B) ∨ C = A ∨ (B ∨ C)

A ∧ B = B ∧ A
A ∨ B = B ∨ A

#### Distributive Laws

A ∧ (B ∨ C) = (A ∧ B) ∨ (A ∧ C)
A ∨ (B ∧ C) = (A ∨ B) ∧ (A ∨ C)

A ∧ (A ∨ B) = A
A ∨ (A ∧ B) = A

#### De Morgan’s Rules

¬(A ∨ B) = ¬A ∧ ¬B
¬(A ∧ B) = ¬A ∨ ¬B

#### Boolean Algebra Practice

Use the formulas listed above to simplify the following Boolean expressions:

#1#2#3#4#5#6
##### Boolean Expression
A ∨ ¬(A ∧ B)

Simplified Boolean Expresssion:

##### Boolean Expression
(A ∧ B) ∨ (A ∧ C)

Simplified Boolean Expresssion:

##### Boolean Expression
(A ∧ B) ∨ A ∧ (B ∨ C)

Simplified Boolean Expresssion:

##### Boolean Expression
¬A ∨ C ∨ (A ∧ B)

Simplified Boolean Expresssion:

##### Boolean Expression
¬(¬A ∧ (B ∧ C))

Simplified Boolean Expresssion:

##### Boolean Expression
¬(A ∧ ¬B) ∨ (¬A ∧ B)

Simplified Boolean Expresssion: