Write a logic expression from a problem statement, logic circuit or truth table

Boolean Logic – IGCSE Computer Science 0478

Key Concepts

Boolean logic uses only two values: TRUE (1) and FALSE (0). The main operators are:

• AND (∧) – true only if both operands are true.
• OR (∨) – true if at least one operand is true.
• NOT (¬) – flips the value.

Example: $A ∧ B$ is true only when both A and B are true.

Analogies

Think of Boolean logic like a traffic light:

• AND = “Both red lights must be on for the car to stop.”
• OR = “Either green light is on, the car can go.”
• NOT = “If the light is red, NOT red means it’s green.”

Writing Expressions from Problem Statements

Suppose the problem is: “A door opens if the alarm is off AND either the key is inserted OR the fingerprint is recognised.”

Let:

• $A$ = alarm is off
• $K$ = key inserted
• $F$ = fingerprint recognised

Expression: $A ∧ (K ∨ F)$

From Logic Circuits to Expressions

Look at the circuit below (described in words):

• Inputs: X, Y, Z
• First gate: AND of X and Y
• Second gate: OR of the AND result with Z

Expression: $(X ∧ Y) ∨ Z$

From Truth Tables to Expressions

Consider this truth table:

From the table we see the output is true when exactly one of A or B is true – that’s an XOR. Expression: $A ⊕ B$ (or equivalently $(A ∧ ¬B) ∨ (¬A ∧ B)$).

Simplification Techniques

1. Use De Morgan’s laws: $¬(A ∧ B) = ¬A ∨ ¬B$
2. Apply the Consensus theorem: $A ∧ B ∨ A ∧ ¬B = A$
3. Combine like terms: $A ∨ A = A$

Practice Problem

Write a Boolean expression for: “A light turns on if the power is on AND (the switch is up OR the motion sensor detects movement).”

Variables:

• $P$ = power is on
• $S$ = switch is up
• $M$ = motion sensor detects movement

Answer: $P ∧ (S ∨ M)$