CLEP College Mathematics: Conditional Logic Practice Questions
Practice translating everyday statements into precise logical claims. These eight original questions cover implications, negation, counterexamples and valid conclusions. Treat each stated rule as a premise, even when a real situation might be more complicated.
Free original practice questions for study purposes. Open Exam Prep is an independent study resource and is not affiliated with, endorsed by, or sponsored by the makers of CLEP.
Answer the questions below — you get instant feedback and a full explanation for each.
1. A rule says, 'If a parcel is fragile, it has a red label.' Which parcel would make this rule false?
Explanation. An implication P implies Q is false exactly when P is true and Q is false. A fragile parcel without the required label satisfies that combination. The rule makes no requirement about nonfragile parcels.
2. Which statement is the contrapositive of 'If an integer is divisible by 12, it is divisible by 3'?
Explanation. For P implies Q, the contrapositive is not Q implies not P: reverse the direction and negate both parts. It has the same truth value as the original implication. Simply reversing the direction gives the converse.
3. Which integer is a counterexample to 'Every even integer is divisible by 4'?
Explanation. A counterexample must satisfy the hypothesis and fail the conclusion. The number 6 is even but is not divisible by 4. The number 9 is not even, so it cannot refute this claim.
4. Let P mean 'the room is open' and Q mean 'the light is on.' Which statement is the negation of 'P and Q'?
Explanation. A conjunction is true only when both parts are true. Its negation is true whenever at least one part is false: not(P and Q) equals (not P) or (not Q). Requiring both parts to be false would be too restrictive.
5. In logic, 'or' is inclusive unless stated otherwise. If P and Q are both true, what is the truth value of 'P or Q'?
Explanation. An inclusive disjunction is true when at least one part is true, including when both are true. Exclusive 'or' would exclude the both-true case, but that is not the operation specified here.
6. Assume this rule is true: 'If a badge is valid, the indicator is green.' The indicator is not green. Which conclusion must follow?
Explanation. From P implies Q and not Q, infer not P. If this badge were valid, the premise would require a green indicator. Nothing in the premises specifies a fault or the color shown by every invalid badge.
7. What is the negation of 'Every file in the folder is encrypted'?
Explanation. A universal claim fails when at least one member fails its condition. Its negation therefore asserts the existence of at least one unencrypted file. It does not require every file, or exactly one file, to be unencrypted.
8. Which expression is logically equivalent to 'If P, then Q' for every assignment of truth values?
Explanation. Both P implies Q and (not P) or Q are false only when P is true and Q is false. Matching all four rows of a truth table establishes logical equivalence.
📘 Looking for more study material? Browse CLEP prep books and study resources. Some links are affiliate links; we may earn a commission at no cost to you.
FAQ
How do I test an implication?
Look for a case in which its hypothesis is true and its conclusion is false. That is the only kind of counterexample.
Does the converse preserve an implication?
Not in general. The contrapositive preserves it; the converse and inverse require separate justification.
What should I practice after these questions?
Draw four-row truth tables for two statements, then translate the same expressions back into ordinary language. This set supports the Logic and Sets portion of CLEP College Mathematics.