1.4.3a Logic Gates & Truth Tables — OCR H446

Boolean Algebra and Logic Gates

Boolean algebra operates on binary values: 1 (TRUE) and 0 (FALSE). Logic gates are the physical hardware implementations of Boolean operations. Every digital circuit — from a simple alarm to a CPU — is built from combinations of these fundamental gates.

The Seven Standard Logic Gates

NOT Gate (Inverter)

Inverts the input. One input, one output. Symbol: ¬A or A̅ or NOT A

A	NOT A (¬A)
0	1
1	0

AND Gate

Output is 1 only when ALL inputs are 1. Symbol: A · B or A AND B

A	B	A AND B
0	0	0
0	1	0
1	0	0
1	1	1

OR Gate

Output is 1 when AT LEAST ONE input is 1. Symbol: A + B or A OR B

A	B	A OR B
0	0	0
0	1	1
1	0	1
1	1	1

NAND Gate (NOT AND)

Output is 0 only when ALL inputs are 1 (inverse of AND). Symbol: ¬(A · B)

A	B	A NAND B
0	0	1
0	1	1
1	0	1
1	1	0

NAND is universal — any logic gate (NOT, AND, OR, XOR, etc.) can be built from NAND gates alone. NAND gates are cheaper to manufacture.

NOR Gate (NOT OR)

Output is 1 only when ALL inputs are 0 (inverse of OR). Symbol: ¬(A + B)

A	B	A NOR B
0	0	1
0	1	0
1	0	0
1	1	0

NOR is also universal — any logic function can be built from NOR gates alone.

XOR Gate (Exclusive OR)

Output is 1 when inputs are DIFFERENT. Symbol: A ⊕ B or A XOR B

A	B	A XOR B
0	0	0
0	1	1
1	0	1
1	1	0

XNOR Gate (Exclusive NOR)

Output is 1 when inputs are THE SAME (inverse of XOR). Symbol: ¬(A ⊕ B)

A	B	A XNOR B
0	0	1
0	1	0
1	0	0
1	1	1

Constructing Truth Tables for Boolean Expressions

A truth table lists all possible input combinations and the corresponding output. For n inputs: 2ⁿ rows.

Example: Complete the truth table for X = (A AND B) OR (NOT C)

A	B	C	A AND B	NOT C	X = (A AND B) OR (NOT C)
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

Method: Work left to right — compute each intermediate column before the final output. Evaluate brackets/NOT first, then AND, then OR.

Logic Gate Symbols (British Standard)

Gate	Symbol description	Boolean notation
NOT	Triangle with bubble at output	¬A, A̅, NOT A
AND	Flat on left, curved D on right	A · B, A AND B
OR	Curved on both sides (like shield)	A + B, A OR B
NAND	AND symbol with bubble at output	¬(A · B)
NOR	OR symbol with bubble at output	¬(A + B)
XOR	OR symbol with extra curved line at input	A ⊕ B
XNOR	XOR symbol with bubble at output	¬(A ⊕ B)

Exam tip: For truth tables with n inputs, you need 2ⁿ rows. Always list inputs in binary counting order (000, 001, 010, 011, 100, 101, 110, 111 for 3 inputs). Add intermediate columns for sub-expressions — you will lose marks if you jump to the answer without working.

Exam tip: NAND and NOR are "universal gates" — this means any other gate (NOT, AND, OR, XOR, XNOR) can be made using only NAND (or only NOR) gates. Examiners often ask you to implement NOT, AND, or OR using NAND gates only: NOT A = A NAND A. A AND B = (A NAND B) NAND (A NAND B).

⚠ Common Mistakes

Confusing XOR with OR — XOR outputs 0 when both inputs are 1 (OR outputs 1). "Exclusive" OR means exclusively one is 1, not both.
Wrong row count — 2 inputs = 4 rows, 3 inputs = 8 rows, 4 inputs = 16 rows.
Not adding intermediate columns — always show A AND B, NOT C etc. as separate columns. Jumping to the output loses working marks.
Confusing NAND (output 0 only when all inputs are 1) with NOR (output 1 only when all inputs are 0).

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Worksheet — 1.4.3a Logic Gates and Truth Tables

8 questions · 20 marks · instantly marked

Q1Complete the full truth table for the expression: X = A OR (NOT B). Use all 4 input combinations.[3 marks]

✓ Mark scheme

A=0,B=0: NOT B=1, X=0 OR 1=1 [½ mark each row]. A=0,B=1: NOT B=0, X=0 OR 0=0. A=1,B=0: NOT B=1, X=1 OR 1=1. A=1,B=1: NOT B=0, X=1 OR 0=1. Correct all 4 rows [2 marks]; correct intermediate NOT B column [1 mark]. Total: [3 marks].

Q2State the output of an AND gate when: (a) A=1, B=0; (b) A=1, B=1; (c) A=0, B=0.[3 marks]

✓ Mark scheme

(a) 0 — AND requires ALL inputs to be 1 [1]. (b) 1 — both inputs are 1 [1]. (c) 0 — neither input is 1 [1].

Q3How does XOR differ from OR? When does XOR output 0 that OR does not?[2 marks]

✓ Mark scheme

XOR outputs 1 only when inputs are DIFFERENT (exactly one input is 1) [1]. When A=1 AND B=1: OR outputs 1, but XOR outputs 0 — this is the only combination where they differ [1]. XOR stands for "exclusive or" — exactly one of the inputs must be 1 (not both).

Q4Complete the full truth table for: X = (A NAND B) with all 4 input combinations.[2 marks]

✓ Mark scheme

A=0,B=0: NAND=1. A=0,B=1: NAND=1. A=1,B=0: NAND=1. A=1,B=1: NAND=0. [1 mark for all correct]; [1 mark for identifying NAND=0 only when A=1 AND B=1]. NAND is the inverse of AND — output is 0 ONLY when all inputs are 1.

Q5How many rows does a truth table with 3 inputs require? List all input combinations in binary counting order.[3 marks]

✓ Mark scheme

2³ = 8 rows [1]. Combinations in order: 000, 001, 010, 011, 100, 101, 110, 111 [2 marks — 1 for all 8 listed, 1 for correct binary counting order]. The pattern: rightmost bit alternates every row, middle bit alternates every 2 rows, leftmost bit alternates every 4 rows.

Q6Explain what it means for NAND to be a "universal gate". Give one example of how to implement a NOT gate using only NAND gates.[3 marks]

✓ Mark scheme

Universal gate: any Boolean logic function / any other gate can be constructed using only NAND gates [1]. This means an entire digital circuit can be built from NAND gates alone — making manufacturing simpler and cheaper [1]. NOT using NAND: connect both inputs of the NAND gate to the same signal A. A NAND A = ¬(A AND A) = ¬A. When A=0: NAND=1; when A=1: NAND=0 [1].

Q7Complete the truth table for the expression X = (A AND B) OR (A XOR C) for A=1, B=0, C=1.[3 marks]

✓ Mark scheme

Step by step: A=1, B=0, C=1. A AND B = 1 AND 0 = 0 [1]. A XOR C = 1 XOR 1 = 0 (inputs are same) [1]. X = 0 OR 0 = 0 [1]. Final answer: X = 0.

Q8Identify two real-world applications of logic gates in digital electronics.[4 marks]

✓ Mark scheme

Any two of [2 marks each]: Arithmetic Logic Unit (ALU) in a CPU — uses combinations of gates (AND, OR, XOR, adders) to perform arithmetic and logical operations [2]. Memory cells (flip-flops) — D-type flip-flops built from NAND gates to store one bit of data in registers and RAM [2]. Alarm systems — AND gates used to combine signals (e.g. sensor=ON AND time=night → activate alarm) [2]. Multiplexers/demultiplexers — used in telecommunications to select/route signals using combinations of AND, OR, NOT gates [2]. Digital comparators — XNOR gates used to compare two binary numbers bit by bit (XNOR outputs 1 when bits match) [2].

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Term	Definition

Boolean Algebra and Logic Gates

The Seven Standard Logic Gates

NOT Gate (Inverter)

AND Gate

OR Gate

NAND Gate (NOT AND)

NOR Gate (NOT OR)

XOR Gate (Exclusive OR)

XNOR Gate (Exclusive NOR)

Constructing Truth Tables for Boolean Expressions

Logic Gate Symbols (British Standard)

Worksheet — 1.4.3a Logic Gates and Truth Tables

Mini Test — 1.4.3a Logic Gates