Boolean Expressions: Study Guide (Computer Science Principles 3.5)

Every decision a program makes comes down to a question with a yes-or-no answer: is the password correct, is the score high enough, has the timer reached zero. In programming the answer is a Boolean value, either true or false. This topic covers how to build and evaluate Boolean expressions with the relational operators (=, ≠, >, <, ≥, ≤) and the logical operators (NOT, AND, OR). Conditionals (Topic 3.6) and loops (Topic 3.8) are driven by these expressions, so accuracy here pays off for the rest of the unit.

Notation. The exam uses its own pseudocode, not a real programming language. The code in this pack is Marlbridge’s rendering of that reference notation, not a copy of the official sheet: ← is assignment, = tests equality, lists start at index 1, DISPLAY shows a value, and braces mark blocks. Use the Exam Reference Sheet from your teacher for the official wording.

Boolean values and relational operators

A Boolean value is either true or false; there is nothing in between. A relational operator compares two values and produces a Boolean:

Operator	Meaning	Example	Value
a = b	equal	4 = 4	true
a ≠ b	not equal	4 ≠ 4	false
a > b	greater than	7 > 3	true
a < b	less than	7 < 3	false
a ≥ b	greater than or equal	5 ≥ 5	true
a ≤ b	less than or equal	6 ≤ 5	false

Two cautions. First, = is a test, not an assignment (Topic 3.1). Second, a comparison is evaluated before it is used: in x ← (5 > 3), the variable x receives the Boolean true.

Logical operators

Logical operators combine Boolean values. Each operand is a Boolean value or a Boolean expression.

p	q	NOT p	p AND q	p OR q
true	true	false	true	true
true	false	false	false	true
false	true	true	false	true
false	false	true	false	false

Read the table as follows.

• NOT p flips the value.
• p AND q is true only when both are true.
• p OR q is true when at least one is true, including when both are. It is inclusive.

Precedence. Many languages evaluate NOT first, then AND, then OR. Do not rely on memory for this: use brackets in your own code so the meaning is obvious, and follow the brackets in exam questions.

Worked example 1: evaluating compound expressions

Let a be 7 and b be 12. Evaluate (i) (a < b) AND NOT(b MOD 5 = 0) and (ii) (a > 10) OR (b = 12 AND a ≠ 7).

a ← 7
b ← 12
DISPLAY((a < b) AND NOT(b MOD 5 = 0))
DISPLAY((a > 10) OR (b = 12 AND a ≠ 7))

Output: true false

(i) Step by step.

Part	Working	Value
a < b	7 < 12	true
b MOD 5	12 MOD 5 = 2	2
b MOD 5 = 0	2 = 0	false
NOT(...)	NOT false	true
whole expression	true AND true	true

(ii) Step by step.

Part	Working	Value
a > 10	7 > 10	false
b = 12	12 = 12	true
a ≠ 7	7 ≠ 7	false
b = 12 AND a ≠ 7	true AND false	false
whole expression	false OR false	false

Check. Evaluate the smallest parts first and work outwards. A single slip, such as reading a ≠ 7 as true, flips the answer, so always write the value of each small comparison.

Worked example 2: from English to Boolean expressions

Task A: “age is between 13 and 19, inclusive”. The expression is age ≥ 13 AND age ≤ 19. Both conditions must hold, so AND is right. A common wrong version uses OR: age ≥ 13 OR age ≤ 19 is true for every age (a 5-year-old satisfies the second part, a 40-year-old the first), so it filters nothing. Test the correct version on four ages:

DISPLAY(5 ≥ 13 AND 5 ≤ 19)
DISPLAY(13 ≥ 13 AND 13 ≤ 19)
DISPLAY(19 ≥ 13 AND 19 ≤ 19)
DISPLAY(40 ≥ 13 AND 40 ≤ 19)

Output: false true true false

The end values 13 and 19 are true, which confirms that ≥ and ≤ include the ends (Figure 1). Using > and < would wrongly exclude them.

Task B: “n is divisible by 3 or by 5, but not by both”. Write the two basic facts, then combine: (n MOD 3 = 0 OR n MOD 5 = 0) AND NOT(n MOD 3 = 0 AND n MOD 5 = 0). Test with 3, 5, 15 and 7:

n	divisible by 3	divisible by 5	expression
3	true	false	true
5	false	true	true
15	true	true	false
7	false	false	false

Fifteen is the interesting case: it satisfies “3 or 5” but also “both”, so the NOT(... AND ...) part removes it.

Task C: equivalent expressions (De Morgan’s laws). Some Boolean expressions can be written in a different but equivalent form. The pair that matters most is

NOT(p AND q) is equivalent to (NOT p) OR (NOT q), and NOT(p OR q) is equivalent to (NOT p) AND (NOT q).

Prove one by truth table:

p	q	p AND q	NOT(p AND q)	NOT p	NOT q	(NOT p) OR (NOT q)
true	true	true	false	false	false	false
true	false	false	true	false	true	true
false	true	false	true	true	false	true
false	false	false	true	true	true	true

The two right-hand columns match in all four rows, so the expressions are equivalent. In words: “it is not true that both are open” means “at least one is closed”.

Common misconceptions

• “OR means exactly one.” p OR q is also true when both are true.
• “age ≥ 13 OR age ≤ 19 means between.” It is true for every age. A range needs AND.
• “≥ and > are interchangeable.” 5 ≥ 5 is true; 5 > 5 is false. Boundary values are where marks are lost.
• “x = 5 OR 6 means x is 5 or 6.” Each side of OR must be a Boolean expression: write x = 5 OR x = 6.
• “NOT(a AND b) is NOT a AND NOT b.” The negation changes AND to OR: NOT(a AND b) equals (NOT a) OR (NOT b).
• “= assigns.” In a Boolean expression = compares. Assignment is ←.

Where this leads

Boolean expressions are the conditions inside IF statements (Topic 3.6, then nested in Topic 3.7), the stopping tests of REPEAT UNTIL loops (Topic 3.8), and the building blocks of equivalent algorithms in Topic 3.9.

Next steps: try the practice questions, skim the revision notes the night before, and tick off the topic checklist. The next topic is Topic 3.6, Conditionals.