In propositional logic , which of the following is equivalent to p → q?
Answer: B
Correct answer: ~p v q
This is the definition of material implication: \( p \to q \) is false only when p is true and q false, which is exactly when \( \lnot p \lor q \) is false. The equivalence is what allows implications to be eliminated from a formula so that De Morgan’s laws and distribution can be applied.
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Fill in the blank
ELFA GLHA ILJA _____ MLNA
Answer: D
Correct answer: KLLA
Take the four positions separately. The first letters go E, G, I, ?, M, advancing by two, so the missing one is K. The second is L throughout and the fourth is A throughout. The third letters go F, H, J, ?, N, also advancing by two, so the missing one is L. That gives KLLA.
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In a top secret message, if LIVING is coded as KGSHLD. How will BUDDHA be coded?
Answer: B
Correct answer: ASACFX
Compare LIVING with KGSHLD letter by letter: L to K is minus one, I to G minus two, V to S minus three, and then the cycle repeats, I to H minus one, N to L minus two, G to D minus three. Applying the same cycle to BUDDHA gives B minus one is A, U minus two is S, D minus three is A, D minus one is C, H minus two is F, and A minus three wraps to X.
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The correct prefix formula is
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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˥(P → Q) is equivalent to
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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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?
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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