Boolean Logic - Truth Tables

Objective

Complete a truth table from a problem statement, logic expression or logic circuit.

Understanding Boolean Logic

Boolean logic deals with logical operations that form the basis of digital circuits and computer programming. The fundamental Boolean values are TRUE (often represented as 1) and FALSE (often represented as 0). We use logical operators to combine and manipulate these values.

Basic Logical Operators

• AND (≈): The output is TRUE only if both inputs are TRUE.
• OR (≨): The output is TRUE if at least one of the inputs is TRUE.
• NOT ( ): The output is the opposite of the input (if the input is TRUE, the output is FALSE, and vice versa).

Constructing Truth Tables

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

To construct a truth table:

1. Identify the number of inputs.
2. List all possible input combinations (each input can be either TRUE or FALSE).
3. For each input combination, evaluate the logical expression and determine the output.

Example 1: Simple AND Expression

Consider the expression: $A \land B$

Here, A and B are the inputs.

The truth table is:

A	B	$A \land B$
FALSE	FALSE	FALSE
FALSE	TRUE	FALSE
TRUE	FALSE	FALSE
TRUE	TRUE	TRUE

Example 2: Simple OR Expression

Consider the expression: $A \lor B$

Here, A and B are the inputs.

The truth table is:

A	B	$A \lor B$
FALSE	FALSE	FALSE
FALSE	TRUE	TRUE
TRUE	FALSE	TRUE
TRUE	TRUE	TRUE

Example 3: NOT Expression

Consider the expression: $\overline{A}$

Here, A is the input.

The truth table is:

A	$\overline{A}$
FALSE	TRUE
TRUE	FALSE

Example 4: Combining Operators

Consider the expression: $(A \land B) \lor \overline{C}$

Here, A, B, and C are the inputs.

The truth table is:

A	B	C	$A \land B$	$\overline{C}$	$(A \land B) \lor \overline{C}$
FALSE	FALSE	FALSE	FALSE	TRUE	TRUE
FALSE	FALSE	TRUE	FALSE	FALSE	FALSE
FALSE	TRUE	FALSE	FALSE	TRUE	TRUE
FALSE	TRUE	TRUE	FALSE	FALSE	FALSE
TRUE	FALSE	FALSE	FALSE	TRUE	TRUE
TRUE	FALSE	TRUE	FALSE	FALSE	FALSE
TRUE	TRUE	FALSE	TRUE	TRUE	TRUE
TRUE	TRUE	TRUE	TRUE	FALSE	TRUE

Practice

Try constructing truth tables for the following expressions:

• $A \lor \overline{B}$
• $\overline{A} \land B$
• $(A \land \overline{B}) \lor \overline{C}$

Suggested diagram: