Boolean Algebra notes

Simplify the Boolean expression: A \cdot (A + B)

Applying the absorption law, the simplified expression is: A

Simplify the Boolean expression: \overline{(A + B)} \cdot A

Applying De Morgan's Law: (\overline{A} \cdot \overline{B}) \cdot A . Rearranging: \overline{A} \cdot A \cdot \overline{B} . Since \overline{A} \cdot A = 0 , the simplified expression is: 0

Simplify the Boolean expression: A \cdot B + A \cdot \overline{B}

Factor out A: A \cdot (B + \overline{B}) . Since B + \overline{B} = 1 , the simplified expression is: A

Simplify the Boolean expression: (A + B) \cdot (A + \overline{B})

Using the distributive law: A \cdot A + A \cdot \overline{B} + B \cdot A + B \cdot \overline{B} . Knowing A \cdot A = A and B \cdot \overline{B} = 0 , we get: A + A \cdot \overline{B} + A \cdot B + 0 . Factor out A: A \cdot (1 + \overline{B} + B) . Since 1 + \text{anything} = 1 , the simplified expression is: A