Truth Tables from Logic Circuits

Syllabus & Goals 3 min

Cambridge 10.1 · Complete a truth table from a logic circuit Paper 2 · Algorithms, Programming and Logic

By the end of this lesson you can:

1. Read a logic circuit, including junction dots and crossing wires.
2. Label the output of each gate and use working columns.
3. Complete an eight-row truth table for a three-input circuit without errors.

Textbook: Chapter 10, §10.3 Example 1 and Activity 10.2 (pp. 360–362, 364–365)

Recap / Warm-Up 5 min

You now know all six gates. Today you join them: the output of one gate becomes the input of the next.

Quick starter

Give the output of each gate when A = 1 and B = 0: (a) NAND (b) NOR (c) XOR.

Key Concept 14 min

1 · Reading a circuit

Signals flow from the inputs on the left to the output on the right. Each gate only ever sees its own two inputs (or one, for NOT). A wire can split to feed two gates; that split is shown with a dot.

2 · The method: one column per gate

1. Count the inputs and write all 2ⁿ rows in counting order. Every row must appear exactly once.
2. Label the output of every gate except the last: P, Q, R… Each label becomes a working column.
3. Fill the columns for gates that only use the original inputs first. Their inputs are already in the table.
4. Then fill the gates fed by P, Q… working left to right. A gate can only be worked out once its inputs are known.
5. Finish with the output X, one whole column at a time. Doing a column at a time keeps you applying one rule, which avoids slips.

3 · Following one row through

Here is one row, A = 1, B = 0, C = 1, traced through the circuit from Worked Example (b). Red wires carry 1; grey wires carry 0.

Worked Example 12 min

(a) A two-input circuit

1. Two inputs, so 4 rows: 00, 01, 10, 11. 2² = 4.
2. P = A AND B: 0, 0, 0, 1. Only row 11 has both inputs at 1.
3. Q = NOT A: 1, 1, 0, 0. Invert column A.
4. X = P OR Q: 1, 1, 0, 1. X is 0 only in row 10, where both P and Q are 0.

(b) A three-input circuit

1. Three inputs, so 8 rows from 000 to 111. 2³ = 8.
2. P = A NAND B. Write 1 in every row except where A = 1 and B = 1 (rows 110 and 111). NAND is 0 only when both inputs are 1.
3. Q = B OR C. Write 0 only where B = 0 and C = 0 (rows 000 and 100). OR is 0 only when both inputs are 0.
4. R = P AND Q. Write 1 only where P and Q are both 1. Use the working columns, not A, B and C.
5. X = R XOR C. Write 1 where R and C are different. C comes straight from the input as well as feeding Q.

Try It Yourself 12 min

📝 Exam Practice 10 min

Recap & Key Terms 3 min

To complete a truth table from a circuit, write every input row, label each gate's output, and fill one working column per gate from left to right. A dot joins wires; a crossing without a dot does not.

Logic circuit

A combination of logic gates designed to carry out a particular task; its output is 0 or 1.

Intermediate value

The output of a gate inside a circuit (e.g. P, Q, R) that feeds another gate.

Working column

An extra truth-table column holding an intermediate value for every row.

Junction dot

A dot where wires join, so the same signal feeds more than one gate.

Homework 1 min

Task (≤ 15 min): Complete the truth table for this circuit. Show a working column for each of the first two gates.