Create and complete truth tables from logic expressions or circuits

Boolean logic is the backbone of digital circuits and computer programs. It uses only two values: True (1) and False (0). In this lesson we’ll learn how to write logic expressions, create truth tables, and translate circuits into tables.

Think of Boolean logic like a recipe that only uses two ingredients: True and False. The “cooking” rules are the logical operators:

• AND ($\land$): both ingredients must be True.
• OR ($\lor$): at least one ingredient is True.
• NOT ($\lnot$): flips the truth value.

Let’s see how each operator works with a simple truth table.

Follow these steps to build a truth table for any Boolean expression:

1. List all input variables (e.g., A, B, C).
2. Determine the number of rows: 2ⁿ where n is the number of variables.
3. Fill in all possible combinations of 0 and 1.
4. Evaluate the expression for each row and write the result.

Example: For the expression (A ∧ B) ∨ ¬C, the truth table would look like this:

Logic gates are the physical representation of Boolean operators:

• AND gate – output is 1 only if all inputs are 1.
• OR gate – output is 1 if at least one input is 1.
• NOT gate – flips the input.

To convert a circuit into a truth table, simply:

1. Identify all input wires.
2. Write the gate logic for each intermediate node.
3. Use the same truth‑table method as above.

Example: A 3‑input NAND gate (¬(A ∧ B ∧ C)) has the following table:

Fill in the missing columns for each expression. Use the same colour scheme as the examples.

1. Expression: A ∨ (B ∧ C)
2. Expression: ¬(A ∨ B) ∧ C
3. Expression: (A ∧ ¬B) ∨ (¬A ∧ B) (XOR)

Know the symbols: ∧ for AND, ∨ for OR, ¬ for NOT.

Always write all input combinations: 2ⁿ rows for n variables.