Compound Boolean Expressions: Revision Notes (Computer Science A 2.5)

Recap

A compound Boolean expression joins Boolean expressions with the logical operators !, && and ||. It always evaluates to true or false.

Expression	True when
!a	a is false
a && b	both a and b are true
a || b	at least one of a, b is true (including both)

Key relationships

Idea	Rule
Precedence	Relational and arithmetic first, then !, then &&, then ||
In a range	x >= lo && x <= hi
Outside a range	x < lo || x > hi
Either of two values	x == 1 || x == 2
Short-circuit &&	If the left side is false, the right side is skipped
Short-circuit ||	If the left side is true, the right side is skipped
Guarding	d != 0 && n / d > 2 is safe; the reverse order is not

Truth table summary

a	b	!a	a && b	a || b
true	true	false	true	true
true	false	false	false	true
false	true	true	false	true
false	false	true	false	false

Assumptions

• Operands of !, && and || are Boolean expressions.
• Short-circuit evaluation applies to && and ||.
• Brackets are used wherever grouping might be misread.

Mistakes to avoid

• Writing x == 1 || 2 or x > 5 || < 3.
• Using || for a range, which gives an always-true condition.
• Using && for “outside the range”, which gives an always-false condition.
• Forgetting that && is evaluated before ||.
• Placing the risky test (division, method call) before its guard.
• Writing !x > 5 instead of !(x > 5).

Worked mini-example

For m = 4 and n = 9, evaluate m > 5 || n > 5 && m + n < 14. Precedence: && before ||. n > 5 is true; m + n is 13, so 13 < 14 is true; true && true is true. m > 5 is false, so the whole expression is false || true, which is true. With brackets (m > 5 || n > 5) && m + n < 14 the answer is the same here (true && true), but for m = 4, n = 12 the first form gives false || (true && 16 < 14), which is false, and the second gives true && false, which is also false. For m = 8, n = 12 the first is true (the left side is true and short-circuits) while the second is true && (20 < 14), which is false. Grouping changes results.

Quick self-check

1. Evaluate true || false && false. Answer: && first gives false; then true || false is true.
2. Write “n is outside 10 to 20 inclusive”. Answer: n < 10 || n > 20.
3. What does false && (5 / 0 > 1) evaluate to without error? Answer: false; the right side is never evaluated.
4. Why is x > 0 || x < 100 a poor range test? Answer: it is true for every x. Use &&.

Full explanations and diagrams are in the Topic 2.5 study guide.