Truth Tables

A truth table is a structured way of showing all possible input combinations for a logic gate or circuit and the resulting outputs. Truth tables are a fundamental tool in GCSE Computer Science — you must be able to read, complete and construct them.

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What Is a Truth Table?

A truth table is a table that lists:

1. Every possible combination of input values (0s and 1s).
2. The output for each combination, according to the logic gate(s) or Boolean expression being evaluated.

Truth tables allow us to see the complete behaviour of a logic system at a glance.

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How Many Rows?

The number of rows in a truth table depends on the number of inputs. For n inputs, there are 2^n (2 to the power of n) possible combinations:

Number of Inputs	Number of Rows (2^n)
1	2
2	4
3	8
4	16

Exam Tip: If a question asks you to draw a truth table for a circuit with 3 inputs, your table MUST have 8 rows of data (plus a header row). Missing rows will lose marks. Use the formula 2^n to check.

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Filling In the Input Columns

There is a systematic way to fill in the input columns so you never miss a combination. For each column, alternate the values in a pattern:

• Rightmost column: alternate every row (0, 1, 0, 1, 0, 1, ...)
• Next column left: alternate every 2 rows (0, 0, 1, 1, 0, 0, 1, 1, ...)
• Next column left: alternate every 4 rows (0, 0, 0, 0, 1, 1, 1, 1, ...)

Example: Three Inputs (A, B, C)

A	B	C
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 equivalent to counting in binary from 000 to 111 (0 to 7 in decimal).

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Single-Gate Truth Tables (Revision)

Here are the truth tables for all six gates you need to know:

NOT Gate (1 input)

A	¬A
0	1
1	0

AND Gate

A	B	A ∧ B
0	0	0
0	1	0
1	0	0
1	1	1

OR Gate

A	B	A ∨ B
0	0	0
0	1	1
1	0	1
1	1	1

XOR Gate

A	B	A ⊕ B
0	0	0
0	1	1
1	0	1
1	1	0

NAND Gate

A	B	¬(A ∧ B)
0	0	1
0	1	1
1	0	1
1	1	0

NOR Gate

A	B	¬(A ∨ B)
0	0	1
0	1	0
1	0	0
1	1	0

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Multi-Gate Truth Tables

In exam questions, you will often need to complete truth tables for circuits or expressions that combine multiple gates. The approach is to add intermediate columns for each gate's output, working from the inputs towards the final output.

Example: A ∧ ¬B

Step 1: List all input combinations for A and B. Step 2: Add an intermediate column for ¬B (NOT B). Step 3: Add the final output column for A ∧ ¬B.

A	B	¬B	A ∧ ¬B
0	0	1	0
0	1	0	0
1	0	1	1
1	1	0	0

Example: (A ∨ B) ∧ ¬C

This expression has three inputs, so the truth table has 2^3 = 8 rows.

A	B	C	A ∨ B	¬C	(A ∨ B) ∧ ¬C
0	0	0	0	1	0
0	0	1	0	0	0
0	1	0	1	1	1
0	1	1	1	0	0
1	0	0	1	1	1
1	0	1	1	0	0
1	1	0	1	1	1
1	1	1	1	0	0

Exam Tip: Always use intermediate columns in your truth table. Even if the question only asks for the final output, showing your working in intermediate columns demonstrates your method and can earn partial marks if you make an error in the final column.

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Working Backwards from a Truth Table

Sometimes an exam question gives you a completed truth table and asks you to identify the gate or write the Boolean expression. To do this:

1. Look at the output pattern.
2. Compare it to the known truth tables for each gate.
3. If it does not match a single gate, consider combinations of gates.

The process from a truth table to a Boolean expression can be drawn as a flow:

graph LR
    TT["Completed truth table"] --> R["Find rows where<br/>output = 1"]
    R --> P["Write a product term<br/>(AND of inputs) per row"]
    P --> S["Sum the product terms<br/>with OR"]
    S --> EXP["Sum-of-products<br/>Boolean expression"]

Quick Identification Guide

Output pattern (for 2 inputs)	Gate
0, 0, 0, 1	AND
0, 1, 1, 1	OR
0, 1, 1, 0	XOR
1, 1, 1, 0	NAND
1, 0, 0, 0	NOR

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Common Mistakes

• Missing rows: Forgetting to include all 2^n combinations.
• Wrong input order: Not listing inputs systematically (use the binary counting method).
• Skipping intermediate steps: Trying to calculate complex expressions in your head instead of using intermediate columns.
• Confusing OR and XOR: Remember that OR gives 1 when both inputs are 1, but XOR gives 0.
• Confusing AND and NAND: NAND is the inverse of AND — every output is flipped.

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Summary

• A truth table shows all possible input combinations and their outputs.
• For n inputs, there are 2^n rows.
• Fill in inputs by counting in binary from all 0s to all 1s.
• Use intermediate columns when evaluating multi-gate expressions.
• You must know the truth tables for NOT, AND, OR, XOR, NAND and NOR.
• Truth tables can be used to identify gates, verify expressions and trace circuits.

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Worked Example: Building a Truth Table From Scratch

Suppose you are asked to construct a truth table for the Boolean expression Q = (A ∧ B) ∨ (¬A ∧ C). Step-by-step procedure:

Step 1 — count inputs. There are three inputs (A, B, C) so the table needs 2^3 = 8 rows.

Step 2 — list inputs in binary order. Use the alternating pattern: C alternates every row, B every two rows, A every four rows.

Step 3 — add intermediate columns. Identify each operator: NOT A, A AND B, (NOT A) AND C, then the final OR.

Step 4 — fill row by row.