Boolean Logic

Objective

Write logic expressions from circuits or truth tables.

Basic Boolean Operators

• AND (Suggested diagram: A circuit with two inputs connected to an AND gate.
  ). Output is 1 only if both inputs are 1. Symbol: Suggested diagram: An AND gate symbol.
  . Logical operator: &
• OR (Suggested diagram: A circuit with two inputs connected to an OR gate.
  ). Output is 1 if at least one input is 1. Symbol: Suggested diagram: An OR gate symbol.
  . Logical operator: |
• NOT (Suggested diagram: A circuit with a single input connected to a NOT gate.
  ). Inverts the input. If the input is 1, the output is 0, and vice versa. Symbol: Suggested diagram: A NOT gate symbol.
  . Logical operator: ←

Truth Tables

A truth table shows all possible combinations of input values and their corresponding output values for a Boolean expression.

Here's an example of a truth table for the expression (A AND B) OR (NOT C):

A	B	C	NOT C	A AND B	(A AND B) OR (NOT C)
0	0	0	1	0	1
0	0	1	0	0	0
0	1	0	1	0	1
0	1	1	0	0	0
1	0	0	1	0	1
1	0	1	0	0	0
1	1	0	1	1	1
1	1	1	0	1	1

Logic Expressions from Circuits

We can translate logic circuits into Boolean expressions.

Consider a circuit with two inputs, A and B, connected to an AND gate, and the output of the AND gate connected to a NOT gate.

The circuit performs the following operation: (A AND B) NOT. The corresponding Boolean expression is: $\overline{A \land B}$ or equivalently $\lnot (A \land B)$

Practice

1. Write the Boolean expression for a circuit with inputs X and Y, an AND gate, and an OR gate with the inputs X and Z.
2. Write the Boolean expression for a circuit with input P and a NOT gate, followed by an AND gate with inputs P and Q.