Logic Gates and Counters

Logic gates combine binary inputs: AND ( $A\cdot B$ ), OR ( $A+B$ ), NOT ( $\bar A$ ), NAND, NOR and XOR. Truth tables list every input combination ( $2^n$ rows). Sequential circuits remember: an astable produces a clock signal, and a binary counter counts its pulses; each stage halves the frequency.

AQA 7408 Paper 3B≈ 40 minRevision 1 · updated 2026-10-27

Section 1 of 14

Learning objectives

Use the six common logic gates and Boolean notation.
Build truth tables for combinations of gates.
Describe binary and BCD counters and frequency division.

Pause and think

A car's interior light comes on if the driver's door OR the passenger door is open. Which gate is that?

Section 2 of 14

Core idea

Start hereDecisions in silicon

"If the door is open AND it's dark, turn on the light." Every such rule can be built from a handful of simple switches called logic gates. Billions of them, clocked millions of times a second, make a computer.

The big idea

Gates combine 0s and 1s; truth tables list all $2^n$ input combinations. Counters chain flip-flops, each halving the clock frequency.

1. Combinational logic

Worked example $Q=\overline{A\cdot B}+C$

For which inputs is $Q=0$ ?

Goal: Rows with $Q=0$ .
Strategy: An OR is 0 only if both terms are 0.

$C=0$
Second term must be 0.
$\overline{A\cdot B}=0\Rightarrow A=B=1$
NAND is 0 only when both inputs are 1.

AnswerOnly

A=1

B=1

C=0

✓ Check7 of 8 rows give

Q=1

✓.

NAND makes NOT and AND

$\overline{A\cdot A}=\bar A$
Join a NAND's inputs: NOT gate.
$\overline{\overline{A\cdot B}}=A\cdot B$
NAND followed by NOT: AND gate.

NAND is universal: every other gate can be built from it.

2. Counters

Worked exampleFrequency division

A $1600$ Hz clock drives a 4-bit binary counter.

Goal: Frequency of each output.
Strategy: Each stage halves.

$f_0=\frac{1600}2=800\text{ Hz}$
Bit 0 toggles once per clock pulse, so it completes a cycle every two pulses.
$800,\ 400,\ 200,\ 100\text{ Hz}$
Each later bit halves again: bits 0 to 3.

AnswerMSB at

100

Hz.

✓ Check

\frac{1600}{2^4}=100

✓.

A counter system

Drawing diagram…

Clock in, binary count out.

Key takeawayBCD A BCD counter resets after 9, so each 4-bit group shows one decimal digit.

Common misconceptionOR vs XOR OR includes the case where both inputs are 1; XOR excludes it.

Pause and think

How many flip-flops are needed to count to 100?

Summary

AND, OR, NOT, NAND, NOR, XOR.
$2^n$ rows.
Counters halve frequency per bit.

Truth table: List of outputs for every input combination.
Astable: Circuit producing a continuous pulse train.
Counter: Sequential circuit counting clock pulses.

HistoryBoole. George Boole, a self-taught mathematician in Ireland, invented the algebra of AND, OR and NOT in 1854, nearly a century before electronic computers.

Fun factBillions of gates. A modern phone processor contains tens of billions of transistors forming logic gates.

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