Logic Questions and Answers

Q: The correct prefix formula is

Correct answer: →→PQ→→QR→PRIn prefix notation the connective precedes its operands, so the string is read outward from the leading symbol. This one parses as an implication whose antecedent is \( P \to Q \) and whose consequent is \( (Q \to R) \to (P \to R) \) - a well-formed formula, and in fact a valid one. The others leave connectives without the operands they require.

Q: ˥(P → Q) is equivalent to

Correct answer: P ^ ˝QAn implication is false in exactly one case, when the antecedent is true and the consequent false, so its negation asserts precisely that case. Formally \( \lnot(P \to Q) \equiv \lnot(\lnot P \lor Q) \equiv P \land \lnot Q \) by De Morgan. Note that the result is a conjunction, not another implication.

Q: A school organized a knockout football competition in which 23 teams played. What was the least number of matches they needed to play to decide the winner?

Correct answer: 22 matchesEvery match eliminates exactly one team, and all but the winner must be eliminated, so 23 teams require 22 matches. The argument is independent of how the byes and rounds are arranged, which is why it gives the answer at once where counting through the rounds would be laborious.

Important Instructions

71.

The correct prefix formula is

A.→˥,P v Q ↔˥R ˥S
B.→P˥Qv ↔ ˥R
C.→→PQ→→QR→PR
D.→˥PV ↔ QSP

Answer: C

Correct answer: →→PQ→→QR→PR

In prefix notation the connective precedes its operands, so the string is read outward from the leading symbol. This one parses as an implication whose antecedent is \( P \to Q \) and whose consequent is \( (Q \to R) \to (P \to R) \) - a well-formed formula, and in fact a valid one. The others leave connectives without the operands they require.

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72.

˥(P → Q) is equivalent to

A.P ^ ˥Q
B.P ^ Q
C.˥P v Q
D.None of these

Answer: A

Correct answer: P ^ ˝Q

An implication is false in exactly one case, when the antecedent is true and the consequent false, so its negation asserts precisely that case. Formally \( \lnot(P \to Q) \equiv \lnot(\lnot P \lor Q) \equiv P \land \lnot Q \) by De Morgan. Note that the result is a conjunction, not another implication.

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73.

A school organized a knockout football competition in which 23 teams played. What was the least number of matches they needed to play to decide the winner?

A.11 matches
B.62 matches
C.21 matches
D.22 matches

Answer: D

Correct answer: 22 matches

Every match eliminates exactly one team, and all but the winner must be eliminated, so 23 teams require 22 matches. The argument is independent of how the byes and rounds are arranged, which is why it gives the answer at once where counting through the rounds would be laborious.

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