Construct a truth table

3.2 Logic Gates and Logic Circuits – Constructing a Truth Table

What is a Truth Table?

A truth table lists every possible combination of input values for a logic circuit and shows the resulting output. Think of it like a recipe card: for each set of ingredients (inputs) you get a specific dish (output).

Common Logic Gates

• AND (∧) – outputs 1 only if all inputs are 1.
• OR (∨) – outputs 1 if at least one input is 1.
• NOT (¬) – flips the input: 0 becomes 1, 1 becomes 0.
• XOR (⊕) – outputs 1 if an odd number of inputs are 1.
• NAND, NOR, XNOR – combinations of the above.

Step‑by‑Step: Building a Truth Table

1. Identify the number of inputs, n. There will be 2ⁿ rows.
2. List all binary combinations from 00…0 to 11…1.
3. For each row, calculate the output using the gate’s Boolean expression.
4. Write the result in the final column.

Example: AND Gate with Two Inputs

Practice: Build a Truth Table for a 3‑Input XOR

Use the steps above. Remember, XOR outputs 1 when an odd number of inputs are 1.

1. Number of rows: 2³ = 8.
2. Write all 8 combinations of A, B, C.
3. Compute A ⊕ B ⊕ C for each row.
4. Fill in the table.

Quick Check: Boolean Expression from a Table

If you have a truth table and want to write the Boolean expression, look for rows where the output is 1. For each such row, write a product (AND) of the inputs (use ¬ for 0). Then OR these products together.

Why Truth Tables Matter

Truth tables let you verify that a circuit does what you expect, debug errors, and design new circuits. They’re the “cheat sheet” for logic designers.

Your Turn! 🎓

• Create a truth table for a 2‑input NAND gate.
• Convert the following table into a Boolean expression.
• Explain in one sentence why a NOT gate is called an inverter.

Summary

You now know how to list every input combination, compute outputs using gate rules, and write the results in a tidy table. Mastering truth tables gives you a solid foundation for all of digital logic.