UGC NET Questions (Paper – 1)

Statements A, B, C and E correctly characterise the biconditional connective. A and C together state that p ↔ q is true when p and q match in truth value and false otherwise. B is correct since p ↔ q decomposes into (p → q) and (q → p). E is true because swapping p and q in a biconditional does not change its meaning. D is false; with two propositional variables, the truth table has four rows, and the biconditional is true in exactly two of them (both true or both false), not three. Therefore, A, B, C and E only form the correct set.

Option A:
Option A is correct because it collects all four accurate properties and excludes D, which miscounts the true rows of the biconditional. It matches the logical behaviour of “if and only if” as tested in NET reasoning.

Option B:
Option B is incomplete as it leaves out E, so it does not mention the commutative nature of the connective, even though that is an important structural property. Without E, the description is not fully comprehensive.

Option C:
Option C is incorrect because, while B and C and E are true, it omits A, which directly states the key condition about matching truth values. The option therefore does not explicitly identify the central truth condition of the biconditional.

Option D:
Option D is wrong because it retains D, the statement that there are three true rows, which is factually incorrect, and it omits B. Once a wrong statement is included, the option cannot represent the exact set of correct statements.

Statements A and B capture that validity is about form, not actual truth of premises. Statement C correctly defines the fallacy of affirming the consequent. Statement E is true because many reasoning questions test the distinction between validity and soundness. Statement D is false since a sound argument is valid and has true premises. Hence A, B, C and E only are correct.

Option A:
Option A is incomplete because it omits E (importance of the distinction) even though A, B, C are true.

Option B:
Option B is incomplete because it omits B (valid with false premises) and also does not include all correct statements together.

Option C:
Option C is correct because it gathers all true statements and excludes D, which reverses the condition for soundness.

Option D:
Option D is wrong because it includes D (false), even though it contains some true statements.

Statements A, B, C, E and F are all correct, while D is false. Logical equivalence requires mutual entailment, so if p and q are equivalent, each entails the other, but a one-way entailment does not guarantee the reverse. Validity of an argument is defined by the impossibility of true premises with a false conclusion, and equivalence between complex forms can be checked via truth tables or algebraic laws. In UGC NET questions, recognising equivalence often allows candidates to prune options or see that two forms say the same thing.

Option A:
Option A is correct because it includes the proper definitions and relationships between equivalence and entailment and connects them to exam strategy, while excluding D, which overgeneralises from entailment to mutual equivalence. This option thus matches standard logical teaching. It also highlights the importance of equivalence in simplifying reasoning tasks.

Option B:
Option B is incomplete since it omits F, failing to mention how recognising equivalence supports option elimination in multiple-choice questions. While A, B, C and E are true, without F the answer does not fully address the UGC NET context. Therefore A, B, C and E only is not the best choice.

Option C:
Option C is wrong because it drops A and includes only B, C, E and F. Without A, the fundamental definition of logical equivalence in terms of truth values is missing, weakening the conceptual foundation. Hence B, C, E and F only cannot be considered complete.

Option D:
Option D is incorrect as it omits B and includes A, C, E and F only. Without B, the definition of argument validity is not stated, which is central to the question. Thus A, C, E and F only does not gather all the correct statements.

A correctly uses ∃! to express unique existence of an additive inverse. D correctly states that “exactly one” implies both existence and uniqueness. B is false (∃! means exactly one). C is false (uniqueness needs a comparison argument). E is false (∀x∃y is not equivalent to ∃y∀x in general). Thus A and D only are correct.

Option A:
Option A is correct as it includes the precise symbolic formulation and the correct meaning of uniqueness, excluding B, C and E which are incorrect.

Option B:
Option B is incorrect because it adds B, which misinterprets ∃! as “infinitely many”.

Option C:
Option C is wrong because it includes E, which is not generally equivalent to ∀x∃y P(x, y).

Option D:
Option D is incorrect because it includes B and E, both of which are false.

Statements A, B, C and D correctly summarise standard relationships on the traditional square of opposition, whereas E is false. Contradictory propositions cannot both be true and cannot both be false, and contraries cannot both be true but may both be false. “All S are P” versus “No S are P” are contraries, while “All S are P” versus “Some S are not P” are contradictories. The pair “Some S are P” and “No S are P” are also contradictories, not contraries, making E the only wrong statement.

Option A:
Option A is incorrect because it includes D as wrong along with E, but D accurately states a classic contradictory pair. Treating both D and E as wrong would misrepresent the structure of the square of opposition.

Option B:
Option B is also wrong since it marks C as wrong along with E, even though C correctly identifies “All S are P” and “No S are P” as contraries. C and E only therefore misclassifies a true statement as false.

Option C:
Option C is correct because it isolates E, the misidentified pair of contraries, as the sole incorrect statement and leaves the accurate descriptions in A, B, C and D untouched. This matches standard logic used in UGC NET preparation.

Option D:
Option D is incorrect as it groups C and D with E, thereby wrongly casting doubt on correctly named contrary and contradictory pairs. C, D and E only thus fails to preserve reliable information about AEIO relations.

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 and D all accurately describe biconditional statements. A gives the overall truth condition: both sides must match in truth value. B shows that “if and only if” can be written as the conjunction of two implications. D restates the falsity condition: the biconditional fails when P and Q differ in truth. C is false because “P or Q” is true in more cases than “P if and only if Q”, and E is false because biconditionals are not tautologies. Hence A, B and D only are correct.

Option A:
Option A is incomplete because it omits D, so it fails to mention explicitly when the biconditional is false, which is essential for constructing its truth table.

Option B:
Option B is incorrect as it includes only B and D and omits A, losing the succinct characterisation that biconditional is true exactly when P and Q agree.

Option C:
Option C is wrong because it adds E, which incorrectly asserts that biconditionals are always true, and it therefore includes at least one false statement.

Option D:
Option D is correct as it brings together the equivalence in terms of matching truth values, the conjunction of implications and the falsity condition, while excluding C and E, which misrepresent the biconditional.

A is correct as a valid categorical syllogism uses exactly three distinct terms: major, minor and middle. B is true because the middle term must be distributed at least once to connect the extremes properly. C is correct since two negative premises provide no positive linkage, and D is true because two particular premises cannot logically support a universal conclusion. E is false because these are necessary but not sufficient conditions; a syllogism can meet them yet still be invalid for other reasons. Hence A, B, C and D only are correct.

Option A:
Option A omits D and thus fails to mention the important constraint on drawing universal conclusions from particular premises. While A, B and C are true, the omission of D makes the set incomplete.

Option B:
Option B ignores A, so it does not mention the three-term requirement; although B, C and D are true, this omission leaves out a central structural rule for syllogisms.

Option C:
Option C drops B and therefore fails to highlight the distribution requirement of the middle term, which is crucial for testing validity. Even with A, C and D present, missing B leaves the account incomplete.

Option D:
Option D is correct because it collects all the true structural rules while excluding E, which overstates their sufficiency. This matches how NET questions test necessary conditions for syllogistic validity.

Statement B gives the correct falsity condition for inclusive disjunction: it fails only when both components are false. C and D are De Morgan’s laws for negation of conjunction and disjunction, both of which are valid equivalences in classical logic. A is false; a conjunction requires both p and q to be true, not just at least one. E is false because in two-valued logic, a statement and its negation cannot be true together. Therefore B, C and D only are correct.

Option A:
Option A is incomplete as it omits D and so does not state the second De Morgan equivalence, leaving the treatment of negation of disjunction unfinished.

Option B:
Option B is incorrect because it includes E, which claims that a statement and its negation can both be true, contradicting the law of non-contradiction in classical logic.

Option C:
Option C is wrong since it accepts A, which incorrectly describes the truth condition for conjunction, and therefore introduces a clear error into the combination.

Option D:
Option D is correct as it gathers the proper truth condition for disjunction and both De Morgan laws, while excluding A and E, which misrepresent conjunction and the mutual exclusivity of a statement and its negation.

Statements A, B and C correctly define converse, inverse and contrapositive for a conditional, and E is true because an implication is logically equivalent to its contrapositive. D is false, as a conditional is not in general equivalent to its converse. Thus the complete set of correct statements about these relationships is A, B, C and E.

Option A:
Option A is incomplete because it omits E, failing to note the important fact that a conditional and its contrapositive are equivalent. Without E, the description of logical relations is not full. Therefore A, B and C only cannot be the correct answer.

Option B:
Option B is correct since it gathers the three accurate definitions and the key equivalence between the original implication and its contrapositive, while excluding D which overstates the relation to the converse. This matches standard treatments of conditionals used in UGC NET.

Option C:
Option C is incorrect because it includes D, which is false (a conditional is not always equivalent to its converse), and it also omits E, which is true. So it mixes a false statement with missing a true one.

Option D:
Option D is incorrect because it omits A, leaving out the explicit definition of the converse, and thus does not list all correct statements. Therefore B, C and E only is not acceptable.

Statement A correctly specifies that the contrapositive of “if p then q” is “if not q then not p”. Statement C is true because a statement and its contrapositive always have the same truth table and are logically equivalent. Statement D is also correct, as the inverse of “if p then q” is “if not p then not q”. Statement B is false because it actually describes the inverse, not the converse; the converse should be “if q then p”. Therefore, the correct set of statements is A, C and D only.

Option A:
Option A is incomplete since it omits D and therefore fails to state the form of the inverse, even though A and C are true. Without D, the picture of how all four related forms are constructed remains partial. This makes the option unsuitable as the correct answer.

Option B:
Option B is incorrect because it excludes C and includes D alone, leaving out the explicit mention of logical equivalence between contrapositives. It does not gather all the correct statements and so cannot represent the full answer.

Option C:
Option C is wrong as it puts together C and D but omits A, so it never explicitly defines the contrapositive itself, while still not distinguishing it from other forms. The missing definition makes this combination insufficient.

Option D:
Option D is correct because it lists the definition of contrapositive, the equivalence of contrapositives and the definition of inverse, while excluding the incorrect wording in B. It therefore gives a coherent and accurate summary of conditional variants.

Statement B correctly negates the universal “all swans are white” by asserting that at least one swan is not white. Statement C is also correct because the negation of “no students are late” is that some students are late. Statement E summarises the general pattern for negating universal statements: replace “all” by “some” and negate the predicate. Statement A is false because “no swans are white” is stronger than the needed negation, and D is false since the negation of “some teachers are not researchers” is “all teachers are researchers”. Therefore, B, C and E only are correct, which matches option B.

Option A:
Option A is incomplete because it omits E (the general rule) and does not include all correct statements needed for the full set.

Option B:
Option B is correct as it combines the proper negations of specific universal and negative statements with the general quantifier rule, while excluding A and D, which overstate or misstate the negations.

Option C:
Option C is wrong because it includes A, which makes the negation of “all swans are white” too strong, so it contains an incorrect statement in the combination.

Option D:
Option D is incorrect since it includes D, which misnegates an existential negative statement, and it omits C, losing one of the correct concrete examples.

Statement A shows a valid chain of universals, so “All poets are creative” follows. C is valid because “No cats are dogs” is equivalent to “No dogs are cats”. E is correct provided at least one engineer exists; then at least one graduate is an engineer. B and D are invalid because they overgeneralise from particular statements. Thus A, C and E only are correct.

Option A:
Option A is incomplete because it omits E, missing the (existence-assumed) inference from “All engineers are graduates” to “Some graduates are engineers”.

Option B:
Option B is wrong because it introduces D, which wrongly generalises from “Some students are not diligent” to “No students are diligent”.

Option C:
Option C is correct because it collects exactly the valid inferences and excludes B and D, which are invalid generalisations.

Option D:
Option D is incorrect because it accepts B, which misreads a particular statement as a universal, and so includes an invalid inference.

Statements A, B and C are all correct. A notes the common natural-language use of “either” as exclusive. B is true because the standard logical disjunction is inclusive and allows both A and B to be true. C correctly gives a formula for exclusive-or using OR, AND and NOT. D is false since exam questions may clarify whether the sense is inclusive or exclusive, and E is false because inclusive and exclusive or differ when both A and B are true. Hence, A, B and C only are correct.

Option A:
Option A is incomplete since it omits C, thereby failing to mention the formal definition of exclusive-or in terms of other connectives, which is important in logical reasoning.

Option B:
Option B is incorrect because it incorrectly includes D, which claims that exam usage is always exclusive, and so introduces a false constraint on interpretation.

Option C:
Option C is correct as it combines natural-language usage, the inclusive logical meaning and the formal construction of exclusive-or, while excluding D and E, which misstate exam conventions and truth-value patterns.

Option D:
Option D is wrong since it includes E, which wrongly claims that inclusive and exclusive or always agree, and it drops A, losing the important point about everyday interpretation.

Statement A pairs a conditional with its contrapositive, so they are logically equivalent. D is correct because logical equivalence by definition means matching truth values across all valuations. E is also true since a conditional and its contrapositive have identical truth conditions, so if the contrapositive is false, the original conditional must be false. B is wrong because its converse is not always true (an even number need not be divisible by 4), and C is wrong because a conditional is not generally equivalent to its converse. Thus A, D and E only form the correct combination.

Option A:
Option A is correct as it lists precisely the statements that capture equivalence with the contrapositive and the notion of equal truth values, while excluding B and C, which overstate the reliability of converses.

Option B:
Option B is incorrect because it adds B, which asserts that the converse of the divisibility statement is always true, even though simple counterexamples like 6 show this is false.

Option C:
Option C is wrong since it brings in C, which claims conditional and converse are always equivalent, contradicting standard logic, so the combination cannot be correct.

Option D:
Option D is incorrect as it omits A and includes B, thereby dropping a correct contrapositive-equivalence example and admitting a false statement about divisibility.