UGC NET Questions (Paper – 1)

Statements A, C and D give standard facts about quantifiers. A correctly identifies “for all n” as a universal quantifier. C is true because the negation of a universal statement is an existential statement with the predicate negated. D is also correct since negating an existential claim yields a universal statement with negated predicate. B is false because no natural number squared equals 2, and E is false because not all natural numbers are even. Therefore A, C and D only are correct.

Option A:
Option A is correct as it lists all the true statements: it recognises the universal quantification in A and the standard negation rules in C and D, while properly excluding B and E, which misstate facts about ℕ.

Option B:
Option B is incomplete since it omits D, ignoring the second important pattern for negating an existential statement. It therefore does not capture the full set of correct statements here.

Option C:
Option C is wrong as it drops A and includes only C and D, failing to mention the basic identification of a universal quantifier in statement A.

Option D:
Option D is incorrect because it adds E, which incorrectly claims that every natural number is even, so the combination contains a false universal statement.

Statements A, B, C, D and E are correct, while F is false. A and B give standard readings of universal and existential quantifiers, and C and D correctly symbolise universal and existential English sentences. E is right that quantifiers operate relative to a chosen domain of objects. F is wrong because formalisation with quantifiers can actually clarify reasoning rather than make it impossible to understand. Thus A, B, C, D and E only is the correct combination.

Option A:
Option A is correct because it collects all the true statements about basic quantifier use and excludes only F, which overstates the difficulty of symbolic form. It accurately reflects how universal and existential claims are treated in predicate logic and in exam questions. Therefore it is the appropriate answer.

Option B:
Option B is incomplete because it omits E, the statement about domain of discourse, which is a fundamental part of quantifier semantics. While A, B, C and D are true, leaving out E makes the description of quantifiers less precise. Hence A, B, C and D only cannot be accepted.

Option C:
Option C is wrong because it includes F among the correct statements. By treating F as true, it implies that quantifiers necessarily obscure reasoning for UGC NET students, which is false. Including this error invalidates the whole combination.

Option D:
Option D is incorrect because it omits B and E and includes F. Without B, the meaning of the existential quantifier is left unstated, and accepting F contradicts the role of formalisation. So A, C, D and F only does not capture the full set of correct statements.

Statements A and B correctly state the meanings of universal and existential quantifiers. Statement D gives the standard rule that the negation of a universal statement becomes an existential statement with negated predicate. Statement E is also true since many mathematical theorems and definitions are framed with such quantifiers. Statement C is false because ∀x P(x) and ∃x P(x) usually have very different truth conditions. Hence A, B, D and E only are correct.

Option A:
Option A is incomplete because it omits E and does not explicitly note the role of quantified statements in formulating general laws and existence results.

Option B:
Option B is wrong because it includes C, the incorrect claim that universal and existential forms are equivalent.

Option C:
Option C is incomplete because it omits A (meaning of ∀) and also does not collect the full correct set together.

Option D:
Option D is correct since it gathers all the true statements and rejects C, which conflates different quantifier types.

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.

Statement C is wrong because the correct negation of “All X are Y” is “Some X are not Y”, not “No X are Y”; denying a universal requires asserting the existence of at least one counterexample, not universal negation. Statements A and B correctly identify universal and existential quantifications, D correctly notes that the negation of “Some X are Y” is “No X are Y”, and E accurately explains the meaning of “Some X are not Y”. Therefore, the only incorrect statement is C, so the option that lists C only is correct.

Option A:
Option A is incorrect because it treats A and C as wrong together, even though A correctly states that “All teachers are researchers” is universally quantified. By marking A as wrong, this option misrepresents basic quantifier classification.

Option B:
Option B is wrong since it pairs C with D as wrong, despite D correctly expressing the negation of an existential statement. Including D among wrong statements contradicts standard quantifier negation rules.

Option C:
Option C is incorrect as it groups B, C and E as wrong, even though B and E correctly explain existential quantification and the meaning of “Some X are not Y”. This option therefore rejects true statements along with the single false one.

Option D:
Option D is correct because it isolates C as the only statement that misstates a quantifier negation, while implicitly affirming the accuracy of A, B, D and E. It preserves the canonical equivalences used in UGC NET logic questions.

Statements A, B, C and D give standard first-order logical translations of everyday sentences. A correctly uses a universal conditional for “all teachers are researchers”. B uses an existential statement with a conjunction for “some students are hardworking”. C appropriately translates “no books are boring” as a universal statement with a negated predicate. D is correct for “some teachers are not researchers” using an existential with a conjunction of teacher and not researcher. E is false because a universal statement does not, by itself, guarantee existence in formal logic, so A, B, C and D only are correct.

Option A:
Option A is incomplete since it omits D and therefore fails to include the important pattern for expressing “some S are not P” in quantifier form. Without D, the range of translation patterns is narrower than NET questions usually expect.

Option B:
Option B is correct because it gathers all four accurate translations and excludes E, which misinterprets what can be inferred from a universal statement alone. It reflects the standard mapping from natural language to quantified logic.

Option C:
Option C is wrong because it leaves out B, ignoring the existential construction for “some students are hardworking”, and still does not address E’s incorrect existential inference. As a result, it is not fully correct.

Option D:
Option D is incorrect since it includes E, which claims existence from a universal statement, an inference that is not valid in general first-order semantics, and also omits A, missing the core universal conditional pattern.

Statements A through E accurately capture how basic quantifier notions are understood and used in reasoning and exams. “At most one” and “exactly one” require structured combinations of quantifiers and identity conditions, as noted in C and D. E is true because Paper 1 often expects conceptual understanding without full symbolism. F is false since the existential quantifier alone cannot express “at most one” or “exactly one” without additional logical resources. Therefore A, B, C, D and E only is the correct combination.

Option A:
Option A is incomplete because it omits E, not acknowledging the UGC NET context in which such quantifier reasoning appears. While A, B, C and D are true, they do not fully answer the question as posed.

Option B:
Option B is correct since it gathers all the true statements about quantifiers and their informal use in exam questions, and it excludes F, which overstates the expressive power of a single quantifier. This matches how quantifier ideas are treated at the Paper 1 level.

Option C:
Option C is wrong as it includes F, suggesting that existential quantification alone can handle “at most one” and “exactly one”. Without identity and sometimes universal conditions, these claims cannot be captured properly. Thus this option is incorrect.

Option D:
Option D is also incorrect because it omits B, losing the basic reading of the universal quantifier, and still fails to include all the correct statements. As such, it cannot be the right choice.