Logical Reasoning & Problem Solving Flashcards

Three conditional statements are given: • "If it rains, the outdoor match is cancelled." • "If the outdoor match is cancelled, the fundraiser fails." • "The fundraiser did not fail." Which conclusion is logically guaranteed?

Answer: It did not rain.

This requires applying the Law of Contrapositive twice in reverse order. 'The fundraiser did not fail' → (contrapositive of statement 2) 'the match was not cancelled' → (contrapositive of statement 1) 'it did not rain.' Options B and C each contradict a step in this valid logical chain. Option D is wrong because the contrapositive chain is a sound deduction.

What is the next term in the following sequence? 1, 4, 9, 61, 52, 63, 94, 46, ?

Answer: 18

Each term is the digit-reversal of a consecutive perfect square: 1²=1→1, 2²=4→4, 3²=9→9, 4²=16→61, 5²=25→52, 6²=36→63, 7²=49→94, 8²=64→46. The next square is 9²=81, and reversing its digits gives 18. Option A (81) is the square itself, not its reversal — the most common trap.

Consider this argument: "Every student who prepared using past papers scored above 70%. Sipho scored above 70%. Therefore, Sipho must have prepared using past papers." This argument is invalid because it commits which formal logical error?

Answer: Affirming the consequent

The argument has the structure: 'If P then Q; Q; therefore P.' This is the fallacy of affirming the consequent. The premise only establishes that past-paper preparation guarantees a score above 70% — not that a high score proves past-paper preparation. Sipho could have studied using textbooks or notes. Denying the antecedent (option A) would look like: 'Sipho did not use past papers, therefore Sipho did not score above 70%.'

In a class of 50 students: 30 study Mathematics, 25 study Physics, and 20 study Chemistry. 15 study both Mathematics and Physics, 10 study both Physics and Chemistry, 12 study both Mathematics and Chemistry, and 5 study all three subjects. How many students study none of these three subjects?

Answer: 7

Apply the inclusion-exclusion principle: |M∪P∪C| = |M|+|P|+|C| − |M∩P| − |P∩C| − |M∩C| + |M∩P∩C| = 30+25+20 − 15−10−12 + 5 = 75 − 37 + 5 = 43. Students studying at least one subject = 43. Students studying none = 50 − 43 = 7. A common error is forgetting to add back the 5 who study all three (subtracted too many times), which gives the wrong answer of 3.

Four professionals — Amara, Bongani, Cyril, and Dineo — each specialise in exactly one area: Audit, Tax, Legal, or IT. The following is known: 1. Amara does not specialise in Audit or Legal. 2. Bongani specialises in IT. 3. Cyril specialises in neither Tax nor IT. 4. Dineo does not specialise in Legal. What does Amara specialise in?

Answer: Tax

From clue 2: Bongani = IT. From clue 1: Amara ≠ Audit and Amara ≠ Legal; since IT is already taken by Bongani, Amara = Tax. Verification: clue 3 eliminates Tax and IT for Cyril, leaving Audit or Legal; clue 4 eliminates Legal for Dineo, so Dineo = Audit and Cyril = Legal. All four assignments are consistent and unique.

A data analyst notes that cities with more fire stations experience more fires annually and concludes: "Fire stations must be causing fires." Which of the following arguments commits the most similar logical error?

Answer: Countries with more doctors per capita report higher illness rates; therefore, doctors are making people sick.

The fire station argument mistakes correlation for causation due to a confounding variable: city size. Larger cities need more fire stations AND have more fires — the stations don't cause the fires. Option B mirrors this exactly: wealthier, larger populations have both more doctors AND more diagnosed illness (better detection), not because doctors cause illness. Option A could reflect genuine causation (nutrition affecting cognition); Option C is an inductive overgeneralisation; Option D is an unwarranted projection from past to future.