Logic Questions and Answers

Answer: D

Correct answer: ((x v y) ↔( ~x v ~y))

This fails whenever x and y have the same truth value: with both true, the left side is true and the right false, so the biconditional is false. The other three are valid - modus ponens, an implication with a self-contradictory antecedent, and the addition law - and a valid formula cannot be the false one.

Answer: A

Correct answer: True

The first equivalence makes a equivalent to \( b \lor \lnot b \), which is a tautology, so a is always true. Now suppose the antecedent \( a \land b \) holds; then b is true, and since \( b \leftrightarrow c \), c is true too, so \( a \land c \) is true and the consequent holds. The implication is therefore true in every case.

Answer: A

Correct answer: True

Take p and q both true. Then \( p \leftrightarrow q \) is true, but \( p \to \lnot q \) is \( T \to F \), which is false. A single row in which the first is true and the second false is enough to show that the first does not imply the second, so the statement that it does not imply it is correct.

Answer: B

Correct answer: He says

A declarative sentence makes a statement and can be assigned a truth value. “It’s right”, “Two may not be an even integer” and “I love you” are all declarative in form. “He says” is a fragment with no complement, asserts nothing and cannot be called true or false, so it is not a statement at all.

Answer: A

Correct answer: JAK

The first letters advance by one through the series: F, G, H, I, so the next is J. The middle letter is A throughout. The last letters run G, F, I, H, so they advance by two and alternate high and low within each pair; the pair beginning with J takes K. The answer is therefore JAK.

Answer: C

Correct answer: 135 chalk sticks

Dividing by half is dividing by one half, which doubles the quantity: \[ \frac{60}{1/2} = 120, \qquad 120 + 15 = 135. \] The trap is to read it as dividing in half, which would give 30 and then 45. The wording is the whole of the question.

Answer: D

Correct answer: Both (a) & (b)

The first is the absorption law: \( (P \lor Q) \land Q \) simplifies to Q, so the biconditional is \( Q \leftrightarrow Q \) and is always true. The second is the disjunctive syllogism: given \( P \lor Q \) and \( \lnot P \), Q must hold. The third fails when P is true and Q false, since the antecedent is then true and the consequent false.

Answer: C

Correct answer: Logically equivalent to p ^ q

Distribute: \( p \land (\lnot p \lor q) \equiv (p \land \lnot p) \lor (p \land q) \), and the first disjunct is a contradiction, so the whole reduces to \( p \land q \). It is therefore neither a tautology nor a contradiction - it is true exactly when both p and q are true.

Answer: B

Correct answer: Father-in-law

Work inward from the description. Smith’s sister’s mother is Smith’s own mother. Her grandson, on Smith’s side of the family, is Smith’s son. Lilly is that son’s wife, so Smith is her husband’s father - her father-in-law.

Answer: A

Correct answer: (P ^ Q) → R

Expand both sides into disjunctive form. The left is \( \lnot P \lor (\lnot Q \lor R) \), which is \( \lnot P \lor \lnot Q \lor R \); the right is \( \lnot(P \land Q) \lor R \), which by De Morgan is the same expression. The result is the exportation law, and it is why a nested implication can be flattened into one with a conjunctive antecedent.