Boolean Logic - Truth Tables

This section focuses on understanding and constructing truth tables for Boolean logic expressions and circuits. Truth tables systematically show the output of a Boolean function for all possible combinations of input values.

Basic Boolean Operators

The fundamental Boolean operators are:

• AND ($ \land $): The output is 1 only if both inputs are 1.
• OR ($ \lor $): The output is 1 if at least one input is 1.
• NOT ($ \neg $): The output is the inverse of the input (1 becomes 0, and 0 becomes 1).

Constructing Truth Tables

To construct a truth table, you need to:

1. Identify the number of inputs.
2. List all possible combinations of input values (each input can be either 0 or 1). The number of rows in the truth table will be $2^n$, where 'n' is the number of inputs.
3. For each row, evaluate the output based on the Boolean operator(s) and the input values.

Example 1: Truth Table for $(A \land B) \lor C$

Let's create a truth table for the expression $(A \land B) \lor C$, where A, B, and C are Boolean variables.

A	B	C	$A \land B$	$(A \land B) \lor C$
0	0	0	0	0
0	0	1	0	1
0	1	0	0	0
0	1	1	0	1
1	0	0	0	0
1	0	1	0	1
1	1	0	1	1
1	1	1	1	1

Example 2: Truth Table for a Simple Circuit

Consider a simple logic gate circuit with two inputs, A and B, connected to an OR gate. The output is the result of the OR operation.

The truth table for this circuit is the same as the truth table for the expression $A \lor B$:

A	B	Output ($A \lor B$)
0	0	0
0	1	1
1	0	1
1	1	1

Completing Truth Tables from Logic Expressions

You can create truth tables for more complex logic expressions by breaking them down into simpler parts and using the properties of Boolean algebra (e.g., De Morgan's Laws, distribution laws) to simplify the expression before constructing the truth table. However, the fundamental principle of listing all possible input combinations remains the same.

Key Takeaways

Truth tables are a fundamental tool in digital logic. They provide a clear and systematic way to understand the behavior of Boolean functions and circuits for all possible input conditions. By constructing and analyzing truth tables, you can determine the logic function represented by a given expression or the output of a logic circuit.