Matching Items MCQs for Sub-Topics of Topic 11: Mathematical Reasoning

(i) A declarative sentence that is either true or false.

(ii) A sentence whose truth value depends on a variable.

(iii) The property of being true or false.

(iv) A sentence that is a command or question.

(v) A sentence that is both true and false simultaneously.

(a) Truth Value

(b) Not a Proposition

(c) Paradoxical Sentence

(d) Proposition

(e) Open Sentence

(i) Proposition (True)

(ii) Proposition (False)

(iii) Open Sentence

(iv) Imperative Sentence

(v) Interrogative Sentence

(a) What is your age?

(b) The capital of Maharashtra is Mumbai.

(c) $x^2 = 9$

(d) $5 + 2 = 8$

(e) Please be quiet.

(i) Definition of a Proposition

(ii) Truth value of "$1+1=3$"

(iii) Open Sentence example

(iv) Truth value of "All prime numbers are odd"

(v) Nature of "This statement is false"

(a) False

(b) Paradox

(c) Declarative and has a definite truth value

(d) True

(e) $x$ is positive

(i) Mathematical Statement

(ii) Non-statement

(iii) True Statement

(iv) False Statement

(v) Sentence with a variable.

(a) Has a definite truth value

(b) Truth value is False

(c) Cannot be assigned True or False

(d) Truth value is True

(e) Open sentence

(i) Are you feeling well?

(ii) Every even number greater than 2 is the sum of two primes (Goldbach's Conjecture).

(iii) $3x - 7 = 5$

(iv) The Earth is a perfect sphere.

(v) Stop the car!

(a) Open Sentence

(b) Proposition with unknown truth value

(c) Interrogative Sentence

(d) Imperative Sentence

(e) Proposition (False)

(i) The door is open.

(ii) All students passed.

(iii) Some apples are red.

(iv) $x = 5$

(v) The number is positive.

(a) Some students did not pass.

(b) The number is not positive (i.e., zero or negative).

(c) The door is not open.

(d) $x \neq 5$

(e) No apples are red (or All apples are not red).

(i) Reverses the truth value of a statement.

(ii) Combines two or more simple statements.

(iii) Symbol for negation.

(iv) Example of a simple statement.

(v) Example of a compound statement.

(a) $\neg$

(b) Negation

(c) The sky is blue and the grass is green.

(d) The Sun is a star.

(e) Compound Statement

(i) $\neg (\neg p)$

(ii) Statement and its negation.

(iii) Negation symbol.

(iv) $\neg (p \land q)$

(v) $\neg (p \lor q)$

(a) $\neg p \land \neg q$

(b) $\sim$

(c) Always false (contradiction).

(d) $\neg p \lor \neg q$

(e) $p$

(i) $p \land q$

(ii) $p \lor q$

(iii) All X are Y.

(iv) Some X are Y.

(v) No X are Y.

(a) Some X are not Y.

(b) Some X are Y.

(c) $\neg p \lor \neg q$

(d) $\neg p \land \neg q$

(e) All X are not Y (or No X are Y).

(i) The number is even and prime.

(ii) The book is interesting or long.

(iii) Every student is present.

(iv) There is a solution.

(v) $a > b$

(a) There is no solution.

(b) Some student is not present.

(c) $a \leq b$

(d) The number is not even or not prime.

(e) The book is not interesting and not long.

(i) $\land$

(ii) $\lor$

(iii) $\neg$

(iv) $p \land q$ truth condition

(v) $p \lor q$ (inclusive) truth condition

(a) True if at least one is true.

(b) Conjunction (AND)

(c) Negation (NOT)

(d) Disjunction (OR)

(e) True only if both are true.

(i) $p$ True, $q$ True

(ii) $p$ True, $q$ False

(iii) $p$ False, $q$ True

(iv) $p$ False, $q$ False

(v) $p$ True

(a) $\neg p$ is False

(b) $p \land q$ is False

(c) $p \lor q$ is True

(d) $p \land q$ is True

(e) $p \lor q$ is False

(i) Number of rows

(ii) Column for $p \land \neg p$

(iii) Column for $p \lor \neg p$

(iv) Exclusive OR ($p \oplus q$) is true

(v) Column for $\neg p$

(a) All False

(b) All True

(c) $2^n$

(d) When $p$ and $q$ have different truth values.

(e) Opposite of $p$'s column.

(i) "and"

(ii) "or"

(iii) "not"

(iv) "neither... nor..." ($p$ and $q$)

(v) "but"

(a) $\land$

(b) $\neg p \land \neg q$

(c) $\lor$

(d) $\neg$

(e) $\land$ (typically)

(i) $p \land q$

(ii) $p \lor q$

(iii) $\neg p \land q$

(iv) $\neg (p \lor q)$

(v) $\neg (\neg p \land \neg q)$

(a) True

(b) False

(c) True (as $\neg p$ is False, $\neg q$ is True, $\neg p \land \neg q$ is False, negation is True)

(d) False (as $p \lor q$ is True, negation is False)

(e) False (as $\neg p$ is False, $\neg p \land q$ is False)

(i) $p \implies q$

(ii) $p \iff q$

(iii) The 'if' part of $p \implies q$

(iv) The 'then' part of $p \implies q$

(v) $p$ is a necessary condition for $q$

(a) Consequent

(b) Antecedent

(c) Conditional Statement

(d) $q \implies p$

(e) Biconditional Statement

(i) $p$ True, $q$ True

(ii) $p$ True, $q$ False

(iii) $p$ False, $q$ True

(iv) $p$ False, $q$ False

(v) $p \implies q$ is false

(a) True

(b) False

(c) Occurs only when $p$ is True and $q$ is False.

(d) True

(e) True

(i) True when antecedent and consequent have the same truth value.

(ii) True only when antecedent is true and consequent is true.

(iii) False only when antecedent is true and consequent is false.

(iv) True when antecedent is true and consequent is false.

(v) True when at least one component is true.

(a) Conjunction ($p \land q$)

(b) $p \implies q$

(c) $p \iff q$

(d) $p \land \neg q$

(e) Disjunction ($p \lor q$)

(i) If you study, then you pass.

(ii) You pass if and only if you study.

(iii) You study only if you pass.

(iv) You pass if you study.

(v) Studying is sufficient for passing.

(a) $p \iff q$

(b) $p \implies q$

(c) $q \implies p$

(d) $p \implies q$

(e) $p \implies q$

(i) Necessary condition for $q$

(ii) Sufficient condition for $q$

(iii) Necessary and sufficient condition for $q$

(iv) Implication

(v) Equivalence

(a) $p \iff q$

(b) $p \implies q$

(c) $p$ in $p \implies q$

(d) $p \implies q$

(e) $p \iff q$

(i) Converse

(ii) Inverse

(iii) Contrapositive

(iv) Original Statement

(v) Negation of the original statement

(a) $\neg p \implies \neg q$

(b) $p \implies q$

(c) $q \implies p$

(d) $p \land \neg q$

(e) $\neg q \implies \neg p$

(i) $p \implies q$ and $q \implies p$

(ii) $p \implies q$ and $\neg p \implies \neg q$

(iii) $p \implies q$ and $\neg q \implies \neg p$

(iv) $q \implies p$ and $\neg p \implies \neg q$

(v) $p \implies q$ and $p \land \neg q$

(a) Inverse pair (logically equivalent)

(b) Contrapositive pair (logically equivalent)

(c) Converse pair (not logically equivalent)

(d) A statement and its negation (logically opposite)

(e) Converse and Inverse (logically equivalent)

(i) Converse

(ii) Inverse

(iii) Contrapositive

(iv) Original Statement

(v) Negation

(a) If a quadrilateral is not a square, then it is not a rhombus.

(b) A quadrilateral is a square and it is not a rhombus.

(c) If a quadrilateral is a rhombus, then it is a square.

(d) If a quadrilateral is a square, then it is a rhombus.

(e) If a quadrilateral is not a rhombus, then it is not a square.

(i) Converse ($q \implies p$)

(ii) Inverse ($\neg p \implies \neg q$)

(iii) Contrapositive ($\neg q \implies \neg p$)

(iv) Negation ($p \land \neg q$)

(v) Biconditional ($p \iff q$)

(a) True

(b) False

(c) May be True or False

(d) True

(e) May be True or False

(i) Logically equivalent to the original.

(ii) Formed by swapping antecedent and consequent.

(iii) Formed by negating antecedent and consequent.

(iv) Formed by swapping and negating antecedent and consequent.

(v) Logically equivalent to the converse.

(a) Inverse ($\neg p \implies \neg q$)

(b) Contrapositive ($\neg q \implies \neg p$)

(c) Converse ($q \implies p$)

(d) Contrapositive ($\neg q \implies \neg p$)

(e) Inverse ($\neg p \implies \neg q$)

(i) $\forall$

(ii) "There exists"

(iii) "Every"

(iv) $\exists$

(v) "For all"

(a) Universal Quantifier

(b) Existential Quantifier

(c) Existential Quantifier

(d) Universal Quantifier

(e) Universal Quantifier

(i) All students are tall.

(ii) Some cars are blue.

(iii) No birds can swim.

(iv) Every dog can bark.

(v) There exists a number $x$ such that $x^2 < 0$.

(a) Some birds can swim.

(b) Some cars are not blue.

(c) There exists a dog that cannot bark.

(d) For every number $x$, $x^2 \geq 0$.

(e) Some students are not tall.

(i) $\forall x, P(x) \implies Q(x)$

(ii) $\exists x, P(x) \land Q(x)$

(iii) $\forall x, P(x) \land Q(x)$

(iv) $\exists x, P(x) \land \neg Q(x)$

(v) $\forall x, P(x)$

(a) All cats are black.

(b) There exists a cat that is not black (Some cats are not black).

(c) Some cats are black.

(d) Everything is a black cat.

(e) Everything is a cat.

(i) Negation of $\forall x, P(x)$

(ii) Negation of $\exists x, P(x)$

(iii) Negation of "All X are Y"

(iv) Negation of "Some X are Y"

(v) Negation of "No X are Y"

(a) Some X are not Y.

(b) All X are not Y (or No X are Y).

(c) $\exists x, \neg P(x)$

(d) Some X are Y.

(e) $\forall x, \neg P(x)$

(i) Statement asserting a property holds for every element.

(ii) Statement asserting a property holds for at least one element.

(iii) Used in proving existence.

(iv) Used in disproving universal statements by counterexample.

(v) Used in proving properties across a domain.

(a) Universal Quantifier ($\forall$)

(b) Existential Quantifier ($\exists$)

(c) Existential Quantifier ($\exists$)

(d) Existential Quantifier ($\exists$)

(e) Universal Quantifier ($\forall$)

(i) Always true.

(ii) Always false.

(iii) Sometimes true, sometimes false.

(iv) $P \iff Q$ is a tautology.

(v) $\neg S$ is a tautology where $S$ is the given statement.

(a) Logical Equivalence

(b) Contradiction

(c) Contingency

(d) Tautology

(e) Contradiction

(i) $p \lor \neg p$

(ii) $p \land \neg p$

(iii) $p \implies p$

(iv) $(p \land q) \lor r$

(v) $\neg (p \implies q) \iff (p \land \neg q)$

(a) Contingency

(b) Tautology

(c) Tautology

(d) Contradiction

(e) Tautology (since $\neg (p \implies q)$ is logically equivalent to $p \land \neg q$, the biconditional is always true)

(i) $\neg (p \land q)$

(ii) $\neg (p \lor q)$

(iii) $p \implies q$

(iv) $\neg (\neg p)$

(v) $p \iff q$

(a) $\neg p \lor \neg q$

(b) $(p \implies q) \land (q \implies p)$

(c) $p$

(d) $\neg p \land \neg q$

(e) $\neg p \lor q$

(i) $p \land (q \lor r) \equiv (p \land q) \lor (p \land r)$

(ii) $p \lor (p \land q) \equiv p$

(iii) $\neg (p \land q) \equiv \neg p \lor \neg q$

(iv) $p \land \text{True} \equiv p$

(v) $p \land q \equiv q \land p$

(a) Commutative Law

(b) Absorption Law

(c) Identity Law

(d) De Morgan's Law

(e) Distributive Law

(i) 1

(ii) 2

(iii) 3

(iv) 4

(v) $n$

(a) 16

(b) 8

(c) 2

(d) 4

(e) $2^n$

(i) Direct Proof of $P \implies Q$

(ii) Proof by Contrapositive of $P \implies Q$

(iii) Proof by Contradiction of $P$

(iv) Proof by Contradiction of $P \implies Q$

(v) Validating a universally quantified statement $\forall x, P(x)$

(a) Assume $\neg (P \implies Q)$ and derive a contradiction.

(b) Assume $P$ and deduce $Q$.

(c) Assume $\neg P$ and derive a contradiction.

(d) Assume $\neg Q$ and deduce $\neg P$.

(e) Prove $P(x)$ for an arbitrary element $x$ in the domain.

(i) Modus Ponens

(ii) Modus Tollens

(iii) Fallacy of the Converse

(iv) Fallacy of the Inverse

(v) A valid argument

(a) $(p \implies q) \land \neg p \implies \neg q$ (Invalid)

(b) $(p \implies q) \land p \implies q$ (Valid)

(c) The conclusion logically follows from the premises.

(d) $(p \implies q) \land \neg q \implies \neg p$ (Valid)

(e) $(p \implies q) \land q \implies p$ (Invalid)

(i) Proving $\forall x, P(x)$

(ii) Disproving $\forall x, P(x)$

(iii) Proving $\exists x, P(x)$

(iv) Disproving $\exists x, P(x)$

(v) Valid argument

(a) Prove $\forall x, \neg P(x)$.

(b) Find at least one $x_0$ such that $\neg P(x_0)$ is true (counterexample).

(c) Show that the premises logically imply the conclusion.

(d) Prove $P(x)$ for a generic $x$ in the domain.

(e) Find at least one $x_0$ such that $P(x_0)$ is true (example).

(i) Valid argument

(ii) Sound argument

(iii) Counterexample

(iv) Role of contradiction in proof by contradiction

(v) Starting point for proof of $P \implies Q$ by contrapositive

(a) A specific case that shows a universal statement is false.

(b) Implies the assumption leading to it must be false.

(c) The premises imply the conclusion.

(d) Assume $\neg Q$.

(e) A valid argument with true premises.

(i) If $n^2$ is odd, then $n$ is odd.

(ii) The sum of any two rational numbers is rational.

(iii) $\sqrt{2}$ is irrational.

(iv) There are infinitely many prime numbers.

(v) If $p \implies q$ and $q \implies r$, then $p \implies r$.

(a) Direct Proof (using properties of rational numbers)

(b) Proof by Contradiction

(c) Proof by Contrapositive (or Direct Proof by considering odd squares)

(d) Proof by Contradiction

(e) Proved using a truth table (or Hypothetical Syllogism)