Logic Questions and Answers

Answer: B

Correct answer: a ^ b → b v c

If the antecedent \( a \land b \) is true then b is true, so \( b \lor c \) is true and the implication holds; if the antecedent is false the implication holds vacuously. The other two fail: the first when a is true and b false, and the third when a is false, b true and c false.

Answer: D

Correct answer: Soles

The question asks what a shoe must have by definition rather than what it commonly has. Laces, leather and straps all appear on some shoes and not others - a slip-on has no laces, a canvas shoe no leather. A sole is what makes the object a shoe at all, so it is the only universal.

Answer: A

Correct answer: History

Chain the preferences together. Social science is preferred to Maths and Maths to History, and English is preferred to Social science, giving English above Social science above Maths above History. Economics is above Maths and Political Science above History. Nothing is placed below History, so it is the least preferred.

Answer: B

Correct answer: Wednesday

Count forward from the fixed point. The day before yesterday was Saturday, so yesterday was Sunday and today is Monday. Tomorrow is Tuesday and the day after tomorrow is Wednesday. The whole difficulty of such questions is keeping the anchor straight, so it is worth naming today explicitly before counting on.

Answer: C

Correct answer: F1 is satisfiable, F2 is valid

F1 is \( P \to \lnot P \), which reduces to \( \lnot P \); that is true when P is false, so F1 is satisfiable though not valid. F2 is \( (P \to \lnot P) \lor (\lnot P \to P) \), which reduces to \( \lnot P \lor P \) and is therefore true in every case, so F2 is valid.

Answer: A

Correct answer: Daughter in law

Suresh and Amar are both sons of Mahesh, being brothers. Jia is married to Amar, so Jia is the wife of one of Mahesh’s sons, which makes her his daughter-in-law. Sia, her sister, is the wife of the other son and stands in exactly the same relation to Mahesh.

Answer: D

Correct answer: Both (b) and (c)

A set is functionally complete if every truth function can be expressed with it. NOR alone will do, since \( \lnot p \) is \( p \downarrow p \) and the rest follows, so the second set is complete; and NAND alone is complete for the same reason, which is why a whole processor can be built from NAND gates. The set with only conjunction and disjunction cannot express negation and so is not complete.

Answer: B

Correct answer: The conjuction F1 ^ F2 is not satisfiable

If both implications are tautologies then whenever \( F_1 \land F_2 \) is true, F3 and its negation are both true - which is impossible. So \( F_1 \land F_2 \) can never be true, and it is unsatisfiable. Nothing follows about F1 or F2 individually; each may be satisfiable on its own.

Answer: C

Correct answer: p v (p→q)

Rewrite the implication: \( p \lor (p \to q) \equiv p \lor \lnot p \lor q \), and \( p \lor \lnot p \) is already true, so the whole is true whatever the values. The other candidate reduces to \( p \lor \lnot q \), which fails when p is false and q true, and the first option fails when p is true and q false.

Answer: C

Correct answer: R

Both P and Q lead to R, and \( P \lor Q \) guarantees that one of them holds, so R follows in either case. This is the constructive dilemma with the same conclusion on both branches, which collapses it to a proof of R. Neither P nor Q individually can be concluded, since the premises do not say which of them is true.