UGC NET Questions (Paper – 1)

Statement E is the only wrong statement in this list. A is correct because induction proofs typically start by verifying the statement for the first natural number in the domain. B is correct since the induction step assumes P(k) and proves P(k+1). C correctly states that if the base and step are valid, the result holds for all relevant n. D is true because induction is a method for proving infinitely many cases using a finite argument. E is false because without proving the base case, there is no guarantee that the chain of implications starts, so the induction proof is incomplete.

Option A:
Option A is incorrect since it claims A alone is wrong, but A accurately describes the base step of induction. Declaring it wrong contradicts standard textbook presentations.

Option B:
Option B is wrong because it treats both A and E as wrong. While E is indeed incorrect, A is a true description of the base case and so cannot be grouped with E as wrong.

Option C:
Option C is correct because it singles out E, which wrongly dismisses the need for a base case, as the only incorrect statement. The other statements together present a coherent picture of mathematical induction.

Option D:
Option D is incorrect because it adds A and C to E in the set of wrong statements. C is a true summary of the outcome of induction, and A correctly describes the base step, so this option misclassifies several true statements as wrong.

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.

This option is correct because the set {0, 1, 2, 3, ...} includes zero and all positive integers. This collection is commonly called the set of whole numbers. It extends the natural numbers by adding zero. Therefore, the notation given describes the whole numbers.

Option A:
Whole numbers consist of zero and all positive counting numbers. The set in the question starts at 0 and continues with 1, 2, 3 and so on, which matches this definition. Hence, whole numbers is the correct term.

Option B:
Integers include negative numbers as well as zero and positive numbers. The set given in the question does not include any negative integers. Thus, the term integers is too broad and does not precisely describe the set.

Option C:
Natural numbers usually refer to positive counting numbers starting from 1, and sometimes definitions vary on including 0. Since the set explicitly lists 0, calling it only natural numbers can be ambiguous. Therefore, this option is not the best fit.

Option D:
Rational numbers include all numbers that can be written as a fraction of integers, including many non-integer values. The set listed contains only special integer values and is much more restricted. Hence, rational numbers is not the correct classification.

In mathematical induction, there are two main parts: the base case and the inductive step. The inductive step assumes that the statement holds for n = k (the induction hypothesis) and then proves that it must also hold for n = k + 1. This step is called the inductive step because it extends the truth from one case to the next. Hence, the description in the question refers to the inductive step.

Option A:
The basis or base step is where we verify that the statement holds for the initial value, usually n = 1 or n = 0. It does not involve the assumption "if true for k then true for k + 1." Therefore, basis does not match the part described in the question.

Option B:
Verification is a general term that could refer to checking any part of the proof. It is not the standard technical name for the specific step that connects n = k to n = k + 1. Hence, it is too vague and not suitable as an answer.

Option C:
The inductive step is correct because it formalises the logical link needed to propagate the truth of the statement from one natural number to the next. Once both the base case and inductive step are established, the statement is proved for all natural numbers in the domain. Recognising this terminology is important for understanding proofs in discrete mathematics and reasoning.

Option D:
Concluding step might describe the final remarks of a proof, but it is not a standard name for the process of moving from k to k + 1. The main logical engine in induction is the inductive step, not a generic conclusion. Thus, this option is not accurate.