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: C

The formula is always valid.

Suppose the antecedent holds, so every object with P also has Q, and suppose there is some object with P. Call it a. Then P(a) is true, so by the first assumption Q(a) is true, and therefore some object has Q.

\[ \begin{aligned} &\forall x(P(x) \to Q(x)),\ P(a) \\ &\Rightarrow Q(a) \Rightarrow \exists x\, Q(x) \end{aligned} \]

The argument uses no property of the domain, so the formula is true in every interpretation, finite or infinite; 'valid only for finite domains' is therefore wrong. Option B confuses the formula with its antecedent: the implication as a whole holds even when P(x) → Q(x) fails for some x, because the implication is then trivially true. ∃x(¬Q(x) ∧ P(x)) is the negation of ∀x(P(x) → Q(x)), not an equivalent of the formula.

Answer: D

A hill always has height.

A hill is by definition a raised landform, higher than the land around it, so elevation is the one feature every hill must have.

Trees, animals and water are often found on hills, but a bare, rocky or dry hill is still a hill, so none of them is essential.

Questions of this kind ask for the feature that belongs to the very definition of the thing, not one that is merely common; height is part of what makes land a hill.

Answer: A

A mirror always reflects.

A mirror is a smooth, highly polished surface whose essential property is to reflect light, so that images are formed. Every mirror, plane, concave or convex, works by reflection.

Refraction, the bending of light as it passes from one medium into another, is how lenses work. Distortion happens only with curved or uneven mirrors, not always, and 'retracts' has nothing to do with light.

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.