Boolean Logic: Gates and Truth Tables

A logic gate is a circuit component that takes one or more binary inputs (0 or 1) and produces a single binary output according to a fixed rule. Digital computers are built from billions of these gates.

The output of each gate is determined entirely by its inputs — there is no analogue gradation. Logic gates are the physical implementation of Boolean algebra, a mathematical system where values are either true (1) or false (0).

AQA 8525 requires knowledge of four gates:

Gate	Inputs	Core rule
NOT	1	Inverts the input
AND	2 or more	Output is 1 only if all inputs are 1
OR	2 or more	Output is 1 if at least one input is 1
XOR	2	Output is 1 if inputs are different

NAND and NOR gates are explicitly excluded from AQA 8525. Do not use them in answers unless the question supplies them.

Logic gates underpin all digital processing — from the arithmetic logic unit (ALU) inside a CPU to the condition checks in every if statement.

The NOT gate (also called an inverter) takes one input and produces its opposite. If the input is 1, the output is 0; if the input is 0, the output is 1.

Boolean notation: $\bar{A}$ (A with an overbar, read "NOT A")

Truth table:

A	$\bar{A}$ (NOT A)
0	1
1	0

The NOT gate is the only single-input gate in AQA 8525. The circuit symbol is a triangle pointing right with a small circle (bubble) at the output.

Worked example — evaluate $\bar{A}$ for both input values:

• A = 0 → NOT gate inverts → output = 1
• A = 1 → NOT gate inverts → output = 0

A truth table for a single-input gate always has exactly 2 rows ($2^1=2$ input combinations). For 2 inputs: 4 rows. For 3 inputs: 8 rows.

The overbar in Boolean algebra ($\bar{A}$) means "the complement of A". It always produces the opposite of A's current value.

AND gate — output is 1 only when all inputs are 1. If any input is 0, the output is 0.

OR gate — output is 1 when at least one input is 1. Output is 0 only when all inputs are 0.

Boolean notation:

• AND: $A\cdot B$ (written A.B)
• OR: $A+B$ (written A+B)

Truth tables (two inputs):

A	B	A.B (AND)	A+B (OR)
0	0	0	0
0	1	0	1
1	0	0	1
1	1	1	1

AND produces a 1 in only one row (both inputs 1). OR produces a 1 in three rows (anything except both 0).

Worked example — evaluate $A\cdot B+\bar{C}$ where A = 1, B = 0, C = 1:

1. $A\cdot B$ = 1 AND 0 = 0
2. $\bar{C}$ = NOT(1) = 0
3. $0+0$ = 0 OR 0 = 0 → final output = 0

AND is the stricter gate — all inputs must be 1. OR is more permissive — just one 1 is enough.

XOR (Exclusive OR) outputs 1 when its inputs are different — one is 0 and the other is 1. When both inputs are the same (both 0 or both 1), the output is 0.

Boolean notation: $A\oplus B$

Truth table:

A	B	$A\oplus B$ (XOR)
0	0	0
0	1	1
1	0	1
1	1	0

XOR differs from OR in exactly one row: when A = 1 and B = 1, OR outputs 1 but XOR outputs 0.

Worked example — evaluate all four input combinations for $A\oplus B$:

• 0 ⊕ 0 = 0 (same → 0)
• 0 ⊕ 1 = 1 (different → 1)
• 1 ⊕ 0 = 1 (different → 1)
• 1 ⊕ 1 = 0 (same → 0)

XOR is used in binary addition (1 + 1 produces a sum bit of 0, which is the XOR of the two inputs), and in simple data checksums and encryption operations.

Mnemonics for XOR: "one but not both". The output is 1 only when exactly one input is 1.

A Boolean expression describes the output of a logic circuit using algebraic notation. The four symbols used in AQA 8525 are:

Operation	Symbol	Example expression
AND	.	$A\cdot B$
OR	+	$A+B$
XOR	⊕	$A\oplus B$
NOT	overbar	$\bar{A}$

Evaluation order: . (AND) has higher precedence than + (OR), just as multiplication has higher precedence than addition. So $A\cdot B+C$ means $(A\cdot B)+C$, not $A\cdot (B+C)$. When the intended order is non-standard, add brackets.

Reading a Boolean expression: evaluate inner operations first, then outer.

Worked example — evaluate $A\cdot B+\bar{C}$ where A = 1, B = 1, C = 0:

1. $\bar{C}$ = NOT(0) = 1
2. $A\cdot B$ = 1 AND 1 = 1
3. $1+1$ = 1 OR 1 = 1 → output = 1

Writing an expression from a circuit: name each gate's output using the correct symbol, working left to right from inputs to final output.

• Gate 1: AND(A, B) → output is $A\cdot B$
• Gate 2: OR($A\cdot B$, C) → final output is $(A\cdot B)+C$

A logic circuit connects gates so the output of one feeds as input into another. For a circuit with three inputs, the truth table has $2^3=8$ rows.

Worked example — circuit: Gate 1 is AND(A, B); Gate 2 is NOT(C); Gate 3 is OR(Gate 1, Gate 2).

Boolean expression for output Z: $Z=(A\cdot B)+\bar{C}$

Complete truth table:

A	B	C	$P=A\cdot B$	$Q=\bar{C}$	$Z=P+Q$
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

Reading the result: Z = 1 whenever C = 0 (because $\bar{C}$ = 1) or when both A = 1 and B = 1. Z = 0 only when C = 1 and at least one of A or B is 0.

Add intermediate columns for each gate output (P, Q above). Building the table one gate at a time eliminates errors from trying to evaluate the whole expression at once.

1. Confusing OR and XOR

OR outputs 1 when one or both inputs are 1. XOR outputs 1 only when exactly one input is 1. When A = 1 and B = 1: OR gives 1, XOR gives 0. This single row distinguishes the two gates.

2. Using NAND or NOR gates

AQA 8525 explicitly excludes NAND and NOR. If a question asks you to construct a logic circuit or Boolean expression, use only NOT, AND, OR, and XOR.

3. Wrong number of rows in the truth table

A circuit with $n$ inputs has exactly $2^n$ rows. Two inputs → 4 rows. Three inputs → 8 rows. Omitting rows or adding extras both lose marks.

4. Misreading the scope of the overbar

The overbar means NOT. Apply it to exactly the part of the expression it covers. If it covers A, it means NOT A. If it covers a bracketed expression, work out the expression first, then invert the result. AQA excludes NAND and NOR as standalone gate symbols, but NOT can still be used as part of a circuit.

5. Precedence errors in Boolean expressions

$A\cdot B+C$ evaluates as $(A\cdot B)+C$, not $A\cdot (B+C)$. The two expressions produce different outputs. When tracing through an expression, resolve . (AND) operations before + (OR) operations.