UGC NET Questions (Paper – 1)

The phrase "for every real number x" indicates that the statement is being made about all real numbers without exception. In logic, this is expressed using the universal quantifier, typically denoted by ∀. It asserts that the property x² ≥ 0 holds for each element in the domain of discourse. Therefore, the statement employs the universal quantifier.

Option A:
Universal quantifier is correct because it captures the idea of generality over an entire set. In symbolic form, the statement would be written as ∀x ∈ ℝ, x² ≥ 0. This form is frequently used in mathematical proofs and theoretical reasoning in aptitude tests and higher mathematics.

Option B:
An existential quantifier is used when we assert that there exists at least one element in the domain with a given property, usually expressed as "there exists." The given statement, however, does not say "there exists some x" but "for every x," making the existential quantifier inappropriate.

Option C:
Negative quantifier is not a standard term in formal logic. Negation can apply to quantified statements, but the main quantifiers are universal and existential. Since the question refers to a well-defined quantifier symbol, this option does not match known terminology.

Option D:
Dual quantifier is not a commonly used name in basic logic courses or in the UGC NET syllabus. While universal and existential quantifiers can be viewed as duals in a theoretical sense, the specific statement here plainly uses the universal form. Thus, this is not the correct label.

A valid argument is one in which, if the premises are true, the conclusion must be true because of the logical structure. Validity concerns the form of the argument rather than the actual truth of the premises. In mathematical reasoning and proofs, ensuring validity guarantees that no incorrect conclusion is drawn from correct assumptions. Therefore, the phrase "conclusion necessarily follows from the premises" directly describes a valid argument.

Option A:
A sound argument is valid and also has all true premises, which is a stronger condition than mere validity. The question, however, focuses only on the logical relation between premises and conclusion, not on whether the premises are true. Because soundness includes an extra requirement not mentioned, this term does not exactly match the description. Hence, it is not the best answer.

Option B:
A strong argument is a term more often used in inductive reasoning to indicate that the premises provide substantial support for the conclusion, but not certainty. Strength suggests high probability, not logical necessity. Since the question emphasises that the conclusion necessarily follows, "strong" is not the correct technical term.

Option C:
A weak argument gives little support to its conclusion and often suffers from logical gaps or insufficient evidence. It certainly does not guarantee that the conclusion follows from the premises. Thus, it is almost the opposite of what the question describes. Consequently, this option is clearly incorrect.

Option D:
Valid is correct because validity captures exactly the idea that the logical form forces the conclusion to follow from the premises. It does not depend on content but on structure, which is crucial in mathematical proofs and reasoning. Recognising validity helps in evaluating arguments on exams and in solving theoretical problems.

For an implication "If P, then Q," the contrapositive is "If not Q, then not P." These two statements are logically equivalent, meaning they are either both true or both false in every situation. The contrapositive reverses the direction of the implication and negates both the hypothesis and the conclusion. Therefore, the given statement "If not Q, then not P" is the contrapositive of "If P, then Q."

Option A:
The converse of "If P, then Q" is "If Q, then P," which merely swaps the positions of P and Q without negating them. It is not logically equivalent to the original statement in general. Since the given statement includes negations, it is not the converse.

Option B:
The inverse of "If P, then Q" is "If not P, then not Q," which negates both parts but keeps the order the same. This is different from "If not Q, then not P," which also reverses the roles of P and Q. Thus, the inverse is not the statement described in the question.

Option C:
The contrapositive is correct because it both negates and reverses the components of the original implication, matching the structure "If not Q, then not P." In logical reasoning, proving the contrapositive is a common technique for establishing the truth of an implication. Recognising this form is important for analysing arguments and proofs.

Option D:
A biconditional statement is of the form "P if and only if Q," signifying that P implies Q and Q implies P simultaneously. It does not involve the pattern "If not Q, then not P" as a standalone statement. Therefore, biconditional is not the correct term here.

This option is correct because the logical negation of a universal statement "All S are P" is an existential statement "Some S are not P." Here, "All students are punctual" negates to "Some students are not punctual." Thus, the correct word to fill the blank is "Some."

Option A:
Saying "No students are not punctual" would mean that every student is punctual, which actually restates the original universal claim in a different form. It does not express the negation. Therefore, this option is incorrect.

Option B:
The word "Most" suggests that a large majority of students are not punctual, which is stronger than merely saying at least one is not punctual. Negation in logic only needs one counterexample, so "most" is not the precise logical opposite of "all."

Option C:
The correct negation of a universal statement is existential, and "some" expresses the existence of at least one counterexample. Saying "Some students are not punctual" states that there is at least one student who fails the property, which contradicts the claim that all are punctual. Thus, this option correctly reflects the logical negation.

Option D:
Saying "Exactly two" students are not punctual again specifies a particular number of exceptions. Logical negation, however, only requires at least one exception and does not fix the number. Hence, this option is not the standard negation.

This option is correct because the symbol '→' represents the logical conditional and is read as "implies." The statement "P → Q" is pronounced "P implies Q." It conveys that whenever P is true, Q must also be true. Therefore, the correct connective word is "implies."

Option A:
The phrase "P causes Q" suggests a causal relationship rather than a logical one. Logical implication does not necessarily assert any physical cause-and-effect link. Thus, "causes" is not the standard reading of the connective.

Option B:
In "P implies Q," "implies" is the accepted wording for the conditional statement. It indicates that truth of P guarantees truth of Q under the logical system. This matches the conventional reading of the symbol '→', so this option is correct.

Option C:
The phrase "P equals Q" refers to equality, which is represented by a different symbol, usually '='. It does not describe a conditional relation where one statement leads to another. Hence, this option is not appropriate.

Option D:
"P contradicts Q" would mean that P and Q cannot both be true, relating to negation rather than implication. That idea is closer to exclusive or or inconsistency, not the ordinary conditional statement. Therefore, this option is incorrect.