1.3.2 Truth Table

A truth table is a way of showing how a logic circuit or Boolean function behaves for every possible combination of inputs. It is one of the most important tools in digital logic.

Definition

A truth table represents a Boolean function or digital logic circuit in table form. It shows how a logic circuit's output or Boolean expression responds to all the possible combinations of the inputs using logic '1' for true and logic '0' for false.

Main Idea

A truth table lists all possible input combinations and shows the corresponding output for each one. It converts the behavior of a Boolean function or digital circuit into a simple table that can be read directly.

Core Concept

Properties of a Truth Table A truth table has the following properties: Truth table consists of rows and columns. It shows the relationship between inputs and output from a Boolean function or digital logic circuit. It shows output for all the possible combinations of inputs using 0 for LOW and 1 for HIGH. All the combinations of inputs are listed in columns on the left, working in the middle, and output is shown in the right most column. The input columns are constructed in the order of binary counting with a number of bits equal to the number of inputs.

Detailed Explanation

Structure of a Truth Table A truth table is organized as follows: Left side input columns (all possible combinations of inputs). Middle working (intermediate steps of the Boolean function). Right most column output.

The input columns follow binary counting. For example, if there are three inputs (x, y, z), the combinations are listed as: 0 0 0 0 0 1 0 1 0 0 1 1 1 0 0 1 0 1 1 1 0 1 1 1 This is the standard binary counting order , with a number of bits equal to the number of inputs. Purpose of the Working Column The working column in the middle of the truth table shows the intermediate results of the Boolean function. This makes it easier to compute and verify the final output, especially for complex expressions. Example: Truth Table for a Boolean Function The Boolean function given in the book is: F=(xyzˉ)+(xyz)+(xˉy)F=(xyzˉ)+(xyz)+(xˉy) The truth table for this function is shown below: x y z Working Output F 0 0 0 1 0

0 0 1 0 0 0 1 0 0 1 0 1 1 0 1 1 0 0 0 0 1 0 1 0 0 1 1 0 1 1 1 1 1 1 1

Important Points

• A truth table represents a Boolean function or digital logic circuit in table form.
• It uses logic '1' for true and logic '0' for false.
• It has rows and columns.
• Input combinations are listed in the left columns.
• The working is shown in the middle.
• The output is shown in the right most column.
• Input columns follow binary counting order , with a number of bits equal to the number of inputs.
• A truth table shows the output for all possible combinations of inputs.

Remember

• Truth table = table form of a Boolean function or digital logic circuit.
• 1 = true, 0 = false.
• Left columns inputs (binary counting order).
• Middle working.
• Right most column output.
• Number of input combinations = 2ⁿ (where n = number of inputs).

Common Mistakes

• Forgetting to list all possible input combinations. A truth table must show every combination.
• Writing input combinations in random order instead of binary counting order .
• Placing the output column in the wrong position. The output must be the right most column.
• Confusing the working column with the output column.
• Using 1 for LOW and 0 for HIGH. The correct convention is 0 for LOW and 1 for HIGH.
• Missing intermediate working steps for complex functions, which makes it hard to verify the output.

Quick Revision

A truth table represents a Boolean function or digital logic circuit in table form. It shows how a logic circuit's output or Boolean expression responds to all possible combinations of the inputs using logic '1' for true and logic '0' for false. A truth table has rows and columns. It shows the relationship between inputs and output. Input combinations are listed in the columns on the left, the working is shown in the middle, and the output is shown in the right most column. The input columns are constructed in the order of binary

counting, with a number of bits equal to the number of inputs. The example from the book shows the truth table for the Boolean function F = (x · y · z̄ ) + (x · y · z) + (x̄ · y), with all eight input combinations and their corresponding outputs.

60-Second Revision

Truth table = Boolean function or digital circuit in table form. Uses 1 for true and 0 for false. Rows and columns. Inputs on the left in binary counting order . Working in the middle. Output in the right most column. Warning: do not skip any input combination — all possibilities must be listed.