Boolean Logic

Boolean logic describes how true/false values combine. Computers use Boolean logic at every level — from transistors in a processor to IF conditions in code. OCR J277 requires knowledge of three logic gates: AND, OR, and NOT.

Each gate takes one or two binary inputs (0 = False, 1 = True) and produces one binary output.

Gate	Symbol name	Inputs	Output is 1 when...
AND	Conjunction	A, B	Both A and B are 1
OR	Disjunction	A, B	At least one of A or B is 1
NOT	Negation	A	A is 0 (inverts the input)

OCR J277 requires AND, OR, and NOT only. XOR, NAND, and NOR are not in the scope of this specification.

Alternative notations accepted in the OCR exam:

• 1/0 may be written as T/F (True/False)
• OR may be written as V (from the Latin vel)
• NOT A may be written as Ā (A with an overbar)

The AND gate outputs 1 only when both inputs are 1. If either input is 0, the output is 0.

Real-world analogy: a security door that requires both a keycard AND a PIN to open.

Truth table for AND:

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

Only the last row outputs 1. Out of 4 combinations, AND outputs 1 for exactly 1.

AND in a logic diagram: the AND gate symbol is a D-shape with a flat back edge. Two input lines enter on the left; one output line exits on the right.

AND in programming:

age = 20
has_id = True
if age >= 18 and has_id:
    print("Entry permitted")

Both conditions must be True for the output to be True — exactly like the AND gate.

The OR gate outputs 1 when at least one input is 1. It outputs 0 only when both inputs are 0.

Real-world analogy: a burglar alarm that triggers if the front door OR the back door is opened.

Truth table for OR:

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

Three out of four combinations output 1. OR outputs 0 only when both inputs are 0.

OR in a logic diagram: the OR gate symbol is a curved D-shape with a concave back edge. Two inputs on the left; one output on the right.

OR in programming:

subject = "Maths"
if subject == "Maths" or subject == "Science":
    print("STEM subject")

The output is True if the student takes Maths or Science (or both).

The NOT gate (inverter) takes a single input and produces the opposite output: 1 becomes 0, and 0 becomes 1.

Real-world analogy: a light that is ON when the switch is NOT pressed.

Truth table for NOT:

A	NOT A
0	1
1	0

NOT in a logic diagram: a triangle (buffer) with a small circle (bubble) on the output line. The bubble indicates inversion.

NOT in programming:

game_over = False
while not game_over:
    playTurn()

The loop continues while game_over is NOT True.

NOT is the only gate with a single input. AND and OR each take exactly two inputs. In a combined diagram, NOT is often placed after another gate to invert its output.

A logic diagram can combine multiple gates to represent complex Boolean expressions. Evaluate from inputs to output, left to right, one gate at a time.

Worked example — evaluate the output of: A AND B, then NOT the result.

Truth table for NOT(A AND B):

A	B	A AND B	NOT(A AND B)
0	0	0	1
0	1	0	1
1	0	0	1
1	1	1	0

Step-by-step for A=1, B=1: AND gate → 1 AND 1 = 1; NOT gate → NOT 1 = 0. Output = 0.

Worked example — evaluate: (A OR B) AND (NOT A)

A	B	A OR B	NOT A	(A OR B) AND (NOT A)
0	0	0	1	0 AND 1 = 0
0	1	1	1	1 AND 1 = 1
1	0	1	0	1 AND 0 = 0
1	1	1	0	1 AND 0 = 0

For each row: compute all intermediate values first, then apply the final gate.

Boolean expressions appear in programming conditions and in exam diagram questions. Being able to convert between a diagram, a truth table, and a written expression is the key skill.

From description to truth table — a sensor system activates an alarm (output = 1) when there is motion (M=1) OR it is dark (D=1) AND an override is NOT active (O=0).

Expression: Output = M OR (D AND NOT O)

M	D	O	NOT O	D AND NOT O	M OR (D AND NOT O)
0	0	0	1	0	0
0	0	1	0	0	0
0	1	0	1	1	1
0	1	1	0	0	0
1	0	0	1	0	1
1	0	1	0	0	1
1	1	0	1	1	1
1	1	1	0	0	1

For 3 inputs there are $2^3=8$ rows. Work column by column from left to right.

1. Confusing AND and OR output rules

AND outputs 1 only when both inputs are 1 — it is the stricter gate.
OR outputs 1 when any input is 1 — it only outputs 0 when both inputs are 0.
Swapping these is the most common error in truth table questions.

2. Missing rows in a truth table

For 2 inputs: 4 rows ($2^2$). For 3 inputs: 8 rows ($2^3$). List inputs in binary counting order (00, 01, 10, 11) to guarantee all combinations are covered.

3. Forgetting to add intermediate columns in combined gates

When a diagram has multiple gates, add a column for each gate's output. Trying to evaluate the final output in one step causes errors. Always work gate by gate.

4. Applying NOT incorrectly

NOT inverts a single value: NOT 1 = 0, NOT 0 = 1. It does not apply to the entire expression unless brackets indicate otherwise. NOT(A AND B) is different from (NOT A) AND (NOT B).

Mistake	Correction
"AND outputs 1 when any input is 1"	That describes OR; AND requires both inputs to be 1
Truth table with 3 rows for 2 inputs	2 inputs → $2^2$ = 4 rows; 3 inputs → $2^3$ = 8 rows
Skipping intermediate gate columns	Add one column per gate; evaluate left to right