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.

The contrapositive of a statement of the form “If p then q” is “If not q then not p,” and these two are logically equivalent. Here, p is “the number is divisible by 6” and q is “the number is divisible by 3.” The contrapositive is therefore “if a number is not divisible by 3, then it is not divisible by 6.” This preserves the logical structure of the original implication while reversing and negating its components.

Option A:
Option A negates p but not q and forms the inverse “If not p then not q,” which is not logically equivalent to the original statement in general.

Option B:
Option B correctly follows the contrapositive pattern ¬q → ¬p, matching the structure of the initial implication about divisibility. It reflects the fact that if a number is not divisible by 3, it cannot possibly be divisible by 6.

Option C:
Option C is the converse q → p, which need not be true simply because the original implication is true; many numbers are divisible by 3 but not by 6.

Option D:
Option D contradicts the original statement directly by combining p with not q, which cannot be true if the original implication holds.