Logic Questions and Answers

Answer: C

The distance between P and R is less than or equal to x + y.

Going from P to R through Q covers x + y km, and a direct route can never be longer than a route through an intermediate point. This is the triangle inequality: PR ≤ PQ + QR. Equality holds only when Q lies on the straight line between P and R.

The distance is not always exactly x + y, because Q need not lie between P and R, and it can never be greater than x + y. Since 'less than or equal to x + y' is always true, 'None of the above' is wrong too.

Answer: C

The aeroplane will be flown by the Child.

An aeroplane is flown by a pilot. In this code, 'pilot' is called 'child', so the person who flies the aeroplane is called child.

The trap is to answer 'Pilot': in the code, 'girl' is called 'pilot', so 'Pilot' would name a girl. 'Girl' in the code means a boy, and 'Man' is not used as a code name for anyone.

Answer: B

The valid conclusion is that it may have rained last night.

The rule is 'if it rains at night, the next morning is cloudy'. A cloudy morning is the consequence, not the cause; mornings can be cloudy for other reasons. Reasoning from the consequence back to the cause ('the morning is cloudy, so it rained') is the fallacy of affirming the consequent, so it cannot be said that it must have rained. But a cloudy morning is consistent with rain, so rain last night remains possible.

The statements say nothing about tonight, so both 'It will rain tonight' and 'It will not rain tonight' are unsupported.

Answer: C

The statement that must be true is Mrs. C has not read the autobiography of Mahatma Gandhi. Mr. A could not have read it either.

Every book A reads, B reads, and every book B reads, C reads. So every book A has read, C has also read. If C has not read a book, A cannot have read it, which is the contrapositive of that chain; the third statement is therefore certain.

The others reverse the direction of the rule. B not reading a book says nothing about C, who may read books B has not. B reading Harry Potter does not mean A did, and C reading all of Coelho does not mean A did, because the rule carries books only from A to B to C, not back.

Answer: D

The three true claims are 1, 4, 6.

Every apple is a fruit, but not every fruit is an apple. So any claim equivalent to ‘if apple, then fruit’ is true, and any claim equivalent to ‘if fruit, then apple’ is false.

• Claim 1 says if apple, then fruit: true.
• Claim 4, ‘fruit if apple’, says the same thing: true.
• Claim 6, ‘not an apple if not a fruit’, is the contrapositive of claim 1: true.
• Claims 2 and 3 both say if fruit, then apple: false. Claim 5, ‘not a fruit if not an apple’, is the contrapositive of that false claim, so it is false too.

Every other option includes at least one of claims 2, 3 or 5.

Answer: D

The statement that follows is India will talk implies the terrorists are handed over.

‘P only if Q’ means that P cannot happen without Q, that is, if P then Q. Here P is 'India will talk' and Q is 'the terrorists are handed over', so talking implies the hand-over has happened.

• 'The terrorists are handed over, hence India will talk' reverses the condition (Q, so P), which 'only if' does not promise.
• 'India will not talk implies that the terrorists have not been handed over' (not P, so not Q) is equivalent to that same reversal.
• 'India will talk though the terrorists are not handed over' directly contradicts the statement.

Answer: D

Correct answer: 3, 5, 1, 4, 2

The sequence follows the natural order of cause and effect. Study comes first, then the exam that tests it, then the job the qualification obtains, then the income the job brings, and finally the independence the income makes possible. Each step is the precondition of the one after it.

Answer: D

Correct answer: Sonam

David is below Dechen and above Sonam, and Karma is above David. So Dechen and Karma are both above David and Sonam is the only one below him. The ordering is not complete - nothing settles whether Dechen or Karma is higher - but the bottom position is determined, and it is Sonam’s.

Answer: C

Correct answer: If p is true and q is false and r is false, the P1 is true

Conjunction binds more tightly than disjunction, so the formula reads \( [(p \lor \lnot q) \land (q \to r)] \lor (r \lor p) \). With p true, q false and r false: \( p \lor \lnot q \) is true, \( q \to r \) is true because the antecedent is false, so the bracket is true and the whole is true. It is not a tautology, since p false, q true and r false makes every disjunct false.

Answer: A

Correct answer: ((P → Q)^(Q → R))→ (P → R)

This is the hypothetical syllogism and it is valid: if P leads to Q and Q leads to R, then P leads to R. The second option is the fallacy of denying the antecedent, the third fails when P is false and Q false, and the fourth confuses the disjunction of two implications with the implication of a disjunction.