UGCNET Questions

For a conditional statement p → q, the contrapositive ¬q → ¬p is logically equivalent to the original. Here p is “the number is divisible by 4” and q is “the number is even.” The contrapositive is “if a number is not even, then it is not divisible by 4.” Both the original statement and its contrapositive are true or false together in all possible cases, so they are logically equivalent.

Option A:
Option A reverses the direction of the implication to q → p, which is the converse. The truth of the original statement does not guarantee the truth of its converse; being even does not ensure divisibility by 4.

Option B:
Option B states “If a number is not divisible by 4, then it is not even,” which is the inverse. The inverse is not logically equivalent to the original because many numbers are not divisible by 4 but are still even.

Option C:
Option C correctly expresses the contrapositive ¬q → ¬p: if a number is not even, then it is not divisible by 4. This preserves the logical content of the original implication and is therefore equivalent to it.

Option D:
Option D, “If a number is not divisible by 4, then it is even,” contradicts the original pattern and misstates the relationship between being even and being divisible by 4.

The contrapositive of a conditional “If P then Q” is “If not Q then not P.” It is logically equivalent to the original statement. Here, P is “it is a bird” and Q is “it can fly.” Negating Q gives “it cannot fly,” and negating P gives “it is not a bird.” Therefore, the contrapositive is “If it cannot fly, then it is not a bird,” which preserves the truth conditions of the original conditional.

Option A:
Option A follows the correct procedure of reversing and negating both the antecedent and consequent. It captures the idea that anything that fails the flight condition cannot belong to the bird category under this rule.

Option B:
Option B is the converse, “If Q then P,” which is not guaranteed to be equivalent; many non birds can fly.

Option C:
Option C is the inverse, “If not P then not Q,” which is also not equivalent and can be false even if the original is true.

Option D:
Option D contradicts the original conditional by stating that non birds can fly as a rule, which is unrelated to the given statement.

Given a conditional “If P then Q,” the inverse is formed by negating both the antecedent and the consequent while keeping their order: “If not P then not Q.” Here, P is “the student is sincere” and Q is “the student passes the exam.” The inverse is therefore “If a student is not sincere, then the student does not pass the exam.”

Option A:
Option A is the converse, reversing the roles of antecedent and consequent to “If Q then P.”

Option B:
Option B correctly negates both antecedent and consequent without swapping them, matching the standard definition of the inverse.

Option C:
Option C negates only the antecedent while leaving the consequent positive, which does not correspond to any standard paired form (it is neither converse, inverse, nor contrapositive of the original).

Option D:
Option D is the contrapositive, “If not Q then not P,” which is logically equivalent to the original conditional, not its inverse.

When we say X is necessary for Y, we mean that Y cannot occur without X. Here, the machine working (Y) cannot happen without electricity (X). In conditional form, that becomes: if the machine works, then electricity is present. This does not say that electricity alone is sufficient to make the machine work; it only says that working implies the presence of electricity.

Option A:
Option A correctly reverses the verbal statement into a conditional where the occurrence of machine functioning guarantees that electricity is available. It respects the asymmetry between necessity and sufficiency.

Option B:
Option B treats electricity as sufficient for the machine to work, which is stronger than what “necessary” states and may not be true if other conditions are required.

Option C:
Option C claims that a failure of the machine implies electricity is present, which is unrelated to the notion of electricity being required for proper functioning.

Option D:
Option D contradicts the original statement by saying the machine works even when the necessary condition, electricity, is absent.