A truth table shows the output of a logical expression for every possible combination of its input variables. It is an essential tool for analysing combinational circuits and verifying that a circuit implements the intended Boolean function.
Why Truth Tables Matter
They provide a systematic way to check the correctness of a logic circuit.
They help in simplifying Boolean expressions using Karnaugh maps or algebraic methods.
They are the basis for designing and testing digital systems such as adders, multiplexers, and encoders.
Steps to Construct a Truth Table
Identify the input variables. List each distinct variable (e.g., \$A\$, \$B\$, \$C\$).
Determine the number of rows. For \$n\$ input variables, the table needs \$2^{n}\$ rows to cover all possible combinations.
Write the binary combinations. Fill the input columns with binary numbers from \$0\$ to \$2^{n}-1\$, usually in ascending order.
Evaluate the expression. For each row, compute the value of intermediate sub‑expressions and the final output.
Record the results. Place the computed values in the appropriate columns of the table.
Example: Constructing a Truth Table for \$F = (A \cdot B) + \overline{C}\$
We will build a truth table for the Boolean function \$F = (A \cdot B) + \overline{C}\$, where \$A\$, \$B\$, and \$C\$ are binary inputs.
\$A\$
\$B\$
\$C\$
\$A \cdot B\$
\$\overline{C}\$
\$F = (A \cdot B) + \overline{C}\$
0
0
0
0
1
1
0
0
1
0
0
0
0
1
0
0
1
1
0
1
1
0
0
0
1
0
0
0
1
1
1
0
1
0
0
0
1
1
0
1
1
1
1
1
1
1
0
1
Notice how the intermediate columns (\$A \cdot B\$ and \$\overline{C}\$) help to verify each step of the calculation before arriving at the final output \$F\$.
Applying the Truth Table to a Logic Circuit
Once the truth table is complete, you can translate it into a physical circuit using basic gates:
\$A \cdot B\$ is implemented with an AND gate.
\$\overline{C}\$ is implemented with a NOT gate.
The final OR operation combines the two results with an OR gate.
Suggested diagram: A circuit showing an AND gate receiving inputs \$A\$ and \$B\$, a NOT gate receiving input \$C\$, and an OR gate combining the outputs of the AND and NOT gates to produce \$F\$.
Practice Exercise
Construct a truth table for the following Boolean expression and then sketch the corresponding circuit:
\$G = \overline{(A + B)} \cdot (B \oplus C)\$
Where \$\oplus\$ denotes the exclusive‑OR (XOR) operation.