Negative Questions MCQs for Sub-Topics of Topic 11: Mathematical Reasoning

Question 1. Which of the following is NOT a mathematical statement?

(A) $2 + 2 = 4$

(B) The capital of India is Delhi.

(C) Is it time for lunch?

(D) Every circle is a square.

Question 2. Which characteristic is NOT required for a sentence to be considered a proposition?

(A) It must be a declarative sentence.

(B) It must be objectively true or false.

(C) It must be related to mathematics.

(D) It cannot be both true and false.

Question 3. Which of the following sentences does NOT have a definite truth value as it is written?

(A) The largest ocean is the Pacific Ocean.

(B) $x$ is an even number.

(C) The smallest prime number is 2.

(D) $10 - 3 = 7$

Question 4. Which statement about the truth value of a proposition is NOT correct?

(A) A proposition is always true or always false.

(B) The truth value is subjective and varies from person to person.

(C) The truth value of "The Sun rises in the west" is False.

(D) Knowing the truth value helps classify a sentence as a proposition.

Question 5. Which of the following is NOT an example of a true mathematical statement?

(A) The sum of the interior angles of a triangle is $180^\circ$.

(B) Every square is a rectangle.

(C) $\sqrt{9} = 3$

(D) All prime numbers are odd.

Question 6. Which type of sentence is generally NOT considered a mathematical statement?

(A) Declarative sentence

(B) Interrogative sentence

(C) Imperative sentence

(D) Exclamatory sentence

Question 7. Which of the following is NOT a proposition because it contains a variable whose value is not specified or quantified?

(A) The number of days in a week is 7.

(B) She is a good student.

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

(D) $2y + 1 = 11$

Question 8. If a sentence is a paradox (like "This statement is false"), why is it NOT considered a mathematical statement?

(A) It is not a declarative sentence.

(B) It contains a variable.

(C) Its truth value cannot be definitively assigned as true or false.

(D) It is a compound statement.

Question 9. Which is NOT a correct way to determine if a sentence is a proposition?

(A) Ask if it makes sense.

(B) Ask if it is a question or a command.

(C) Ask if it can be assigned a truth value of true or false.

(D) Ask if it is a statement of fact.

Question 10. Which statement about identifying mathematical statements is NOT accurate?

(A) All mathematical equations are propositions.

(B) Opinions are generally not propositions.

(C) Commands are not propositions.

(D) Questions are not propositions.

Question 1. If $p$ is the statement "The weather is warm", which of the following is NOT a correct way to state the negation of $p$?

(A) It is not warm.

(B) The weather is cold.

(C) It is false that the weather is warm.

(D) It is not true that the weather is warm.

Question 2. Which of the following statements is NOT the correct negation of "Some students passed the exam"?

(A) Some students did not pass the exam.

(B) No student passed the exam.

(C) All students failed the exam.

(D) Every student did not pass the exam.

Question 3. Which of the following is NOT a compound statement?

(A) It is raining and I will stay home.

(B) The number is prime or even.

(C) The cat sat on the mat.

(D) If you study, then you will pass.

Question 4. If a simple statement $p$ is false, which of the following statements is NOT true?

(A) $\neg p$ is true.

(B) $\neg (\neg p)$ is false.

(C) $p \land q$ is false (for any statement $q$).

(D) $p \lor q$ is true (for any statement $q$).

Question 5. Which symbol is NOT typically used to represent negation in logic?

(A) $\neg$

(B) $\sim$

(C) - (as in -p)

(D) $\land$

Question 6. Which is NOT a logical connective used to form compound statements?

(A) AND

(B) OR

(C) IS

(D) NOT

Question 7. The negation of "The number is positive and even" is NOT:

(A) The number is not positive or not even.

(B) The number is negative or odd.

(C) It is false that (the number is positive and even).

(D) The number is not positive AND not even.

Question 8. If a compound statement is formed using the connective "or", it is called a:

(A) Conjunction

(B) Disjunction

(C) Negation

(D) Conditional

Question 9. Which statement about the symbol $\lor$ is NOT correct?

(A) It represents the logical connective OR.

(B) It is used to form disjunctions.

(C) $p \lor q$ is false only when $p$ and $q$ are both false (inclusive OR).

(D) It represents the logical connective AND.

Question 10. If $p$ is false and $q$ is true, which of the following compound statements is NOT true?

(A) $\neg p$

(B) $p \lor q$

(C) $p \land q$

(D) $\neg p \land q$

Question 1. Which statement about the truth table for conjunction ($p \land q$) is NOT correct?

(A) It has 4 rows if $p$ and $q$ are the only simple statements.

(B) The column for $p \land q$ contains 'T' only in the first row (where $p$ and $q$ are T).

(C) The column for $p \land q$ contains 'F' in all rows except the first.

(D) The column for $p \land q$ is identical to the column for $p \lor q$.

Question 2. Which statement about the truth table for inclusive disjunction ($p \lor q$) is NOT correct?

(A) It has 4 rows if $p$ and $q$ are the only simple statements.

(B) The column for $p \lor q$ contains 'F' only in the last row (where $p$ and $q$ are F).

(C) The column for $p \lor q$ contains 'T' in all rows except the last.

(D) The column for $p \lor q$ contains 'T' only when $p$ is true and $q$ is true.

Question 3. Which statement about the truth table for negation ($\neg p$) is NOT correct?

(A) It has 2 rows if $p$ is the only simple statement.

(B) The column for $\neg p$ is the opposite of the column for $p$.

(C) If the column for $p$ is T, F, then the column for $\neg p$ is F, T.

(D) If $p$ is a tautology, then $\neg p$ is a tautology.

Question 4. How many rows are NOT needed in the truth table for a compound statement involving $n$ distinct simple propositions?

(A) $2^n - 1$

(B) $2^n - 2$

(C) $n^2 - 1$

(D) Any number of rows other than $2^n$.

Question 5. If $p$ is true and $q$ is true, which statement is NOT correct?

(A) $p \land q$ is true.

(B) $p \lor q$ is true.

(C) $\neg p$ is false.

(D) $\neg q$ is true.

Question 6. If $p$ is false and $q$ is false, which statement is NOT correct?

(A) $p \land q$ is false.

(B) $p \lor q$ is false.

(C) $\neg p$ is false.

(D) $\neg q$ is true.

Question 7. Which of the following connectives is NOT a binary connective (connecting two statements)?

(A) Conjunction ($\land$)

(B) Disjunction ($\lor$)

(C) Negation ($\neg$)

(D) Biconditional ($\iff$)

Question 8. When constructing a truth table for $\neg (p \land q)$, which intermediate column is NOT typically needed?

(A) Column for $p$

(B) Column for $q$

(C) Column for $p \land q$

(D) Column for $p \lor q$

Question 9. If $p$ is false, which of the following statements must NOT be true?

(A) $\neg p$

(B) $p \lor q$ (if $q$ is true)

(C) $p \land q$ (if $q$ is true)

(D) $\neg p \lor q$ (if $q$ is false)

Question 10. Which of the following represents an exclusive OR between $p$ and $q$?

(A) $p \lor q$

(B) $p \oplus q$

(C) True when $p$ and $q$ have different truth values.

(D) True when $p$ and $q$ have the same truth value.

Question 1. Which of the following is NOT an equivalent way to express the conditional statement "$p \implies q$"?

(A) If $p$, then $q$.

(B) $q$ if $p$.

(C) $p$ only if $q$.

(D) $p$ is a necessary condition for $q$.

Question 2. The conditional statement $p \implies q$ is NOT true when:

(A) $p$ is true and $q$ is true.

(B) $p$ is false and $q$ is true.

(C) $p$ is true and $q$ is false.

(D) $p$ is false and $q$ is false.

Question 3. Which of the following statements about the biconditional ($p \iff q$) is NOT correct?

(A) It is read as "$p$ if and only if $q$".

(B) It is logically equivalent to $(p \implies q) \lor (q \implies p)$.

(C) It is true when $p$ and $q$ have the same truth value.

(D) It is false when $p$ and $q$ have different truth values.

Question 4. In the statement "If you get good marks (g), then you studied hard (s)", which part is NOT correctly identified?

(A) $g$ is the antecedent.

(B) $s$ is the consequent.

(C) The logical form is $g \implies s$.

(D) Getting good marks is a necessary condition for studying hard.

Question 5. If $p \implies q$ is true and $p$ is false, which of the following is NOT a possible truth value for $q$?

(A) True

(B) False

(C) Cannot be determined from the given information.

(D) Both True and False are possible for $q$.

Question 6. If $p \iff q$ is false, which statement is NOT correct?

(A) $p$ is true and $q$ is false.

(B) $p$ is false and $q$ is true.

(C) $p$ and $q$ have the same truth value.

(D) $p$ and $q$ have different truth values.

Question 7. Which of the following is NOT a symbol used for the conditional statement?

(A) $\implies$

(B) $\to$

(C) $\subset$

(D) $\therefore$

Question 8. The statement "$q$ is necessary for $p$" is NOT equivalent to:

(A) $p \implies q$

(B) If $p$, then $q$.

(C) If not $q$, then not $p$.

(D) $q$ if $p$.

Question 9. Which row in the truth table makes the statement $p \implies q$ NOT true?

(A) T, T

(B) T, F

(C) F, T

(D) F, F

Question 10. If $p \iff q$ is true, which of the following is NOT necessarily true?

(A) $p \implies q$ is true.

(B) $q \implies p$ is true.

(C) $\neg p \iff \neg q$ is true.

(D) $p$ is true and $q$ is true.

Question 1. Given the conditional statement $p \implies q$, which of the following is NOT a related conditional statement?

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

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

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

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

Question 2. Which pair of statements is NOT logically equivalent?

(A) Conditional and Contrapositive

(B) Converse and Inverse

(C) Conditional and Inverse

(D) Conditional ($p \implies q$) and $\neg p \lor q$

Question 3. If the original statement "$p \implies q$" is true, which of the following is NOT necessarily true?

(A) The contrapositive ($\neg q \implies \neg p$)

(B) The converse ($q \implies p$)

(C) The inverse ($\neg p \implies \neg q$)

(D) If $p$ is true, $q$ is true.

Question 4. Given the statement "If a number is prime (p), then it is odd (q)". Which of the following related statements is NOT true?

(A) The original statement is false (because 2 is prime and even).

(B) The converse ("If a number is odd, then it is prime") is false (e.g., 9 is odd but not prime).

(C) The inverse ("If a number is not prime, then it is not odd") is true.

(D) The contrapositive ("If a number is not odd, then it is not prime") is true (If even, then not prime, because only 2 is even and prime).

Question 5. How is the inverse of a conditional statement $p \implies q$ NOT formed?

(A) By negating the antecedent and the consequent.

(B) By swapping the antecedent and the consequent and then negating both.

(C) By taking the contrapositive of the converse.

(D) Symbolically as $\neg p \implies \neg q$.

Question 6. If the inverse of a conditional statement is false, which of the following is NOT necessarily true?

(A) The converse is false.

(B) The original conditional statement is true.

(C) The contrapositive is true.

(D) The negation of the original statement is true.

Question 7. Which statement about related conditional statements is NOT correct?

(A) A conditional statement and its contrapositive are logically equivalent.

(B) The converse and the inverse of a conditional statement are logically equivalent.

(C) The negation of $p \implies q$ is logically equivalent to $\neg p \implies \neg q$.

(D) The negation of $p \implies q$ is logically equivalent to $p \land \neg q$.

Question 8. The contrapositive of "If it snows, then the roads are slippery". Which of the following is NOT the contrapositive?

(A) If the roads are not slippery, then it does not snow.

(B) If it does not snow, then the roads are not slippery.

(C) The roads are not slippery if it does not snow.

(D) It is false that (it snows and the roads are not slippery).

Question 9. If the converse of a statement is false, which of the following MUST be true?

(A) The original statement is false.

(B) The inverse is false.

(C) The contrapositive of the original statement is false.

(D) The original statement is true.

Question 10. Which statement is NOT equivalent to $p \implies q$?

(A) $\neg p \lor q$

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

(C) $p \land \neg q$

(D) It is not the case that ($p$ is true and $q$ is false).

Question 1. Which phrase does NOT indicate the use of a universal quantifier?

(A) All

(B) Every

(C) For each

(D) Some

Question 2. Which symbol does NOT represent a quantifier?

(A) $\forall$

(B) $\exists$

(C) $\neg$

(D) There exists

Question 3. Which is NOT the correct negation of the statement "All students are present"?

(A) Some students are not present.

(B) Not all students are present.

(C) There exists a student who is not present.

(D) All students are absent.

Question 4. Which statement about the existential quantifier ($\exists$) is NOT correct?

(A) It is read as "there exists".

(B) It means the property holds for every element.

(C) It means the property holds for at least one element.

(D) It is used in statements like "Some numbers are even".

Question 5. The negation of "There exists a solution" is NOT equivalent to:

(A) There is no solution.

(B) For every possible value, it is not a solution.

(C) All values are not solutions.

(D) There are some values that are not solutions.

Question 6. Which statement uses a quantifier implicitly?

(A) The Sun is a star.

(B) $x^2 = 4$

(C) All dogs bark.

(D) Some cats meow.

Question 7. The negation of a statement involving the universal quantifier ($\forall x, P(x)$) is NOT:

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

(B) Some $x$ such that $P(x)$ is false.

(C) Not all $x$ satisfy $P(x)$.

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

Question 8. Which statement does NOT involve a quantifier?

(A) Every integer is positive.

(B) There exists a number greater than 100.

(C) If it is raining, the ground is wet.

(D) Some students are absent.

Question 9. Which statement about negating quantified statements is NOT correct?

(A) The negation of a universal statement is an existential statement.

(B) The negation of an existential statement is a universal statement.

(C) When negating a quantified statement, you negate the quantifier and the predicate.

(D) The negation of "All P are Q" is "No P are Q".

Question 10. Let $P(x)$ be "$x$ is a cat" and $Q(x)$ be "$x$ can talk". Which symbolic statement represents "No cat can talk"?

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

(B) $\forall x, P(x) \implies \neg Q(x)$

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

(D) $\forall x, P(x) \land \neg Q(x)$

Question 1. Which property is NOT true for a tautology?

(A) Its truth table column contains only 'True' values.

(B) It is logically equivalent to the statement "$\text{True}$".

(C) Its negation is a contradiction.

(D) It is logically equivalent to $p \land \neg p$ for any statement $p$.

Question 2. Which property is NOT true for a contradiction?

(A) Its truth table column contains only 'False' values.

(B) It is logically equivalent to the statement "$\text{False}$".

(C) Its negation is a tautology.

(D) It is logically equivalent to $p \lor \neg p$ for any statement $p$.

Question 3. A contingency is a statement that is NOT:

(A) Always true.

(B) Always false.

(C) A tautology.

(D) A contradiction.

Question 4. Two statements $P$ and $Q$ are logically equivalent. Which statement is NOT true about them?

(A) They have the same truth value in all cases.

(B) The biconditional $P \iff Q$ is a tautology.

(C) The conditional $P \implies Q$ is a tautology.

(D) Their truth table columns are identical.

Question 5. When constructing a truth table for $(p \land q) \lor r$, which step is NOT necessary?

(A) Listing all possible truth values for $p, q, r$.

(B) Creating a column for $p \land q$.

(C) Creating a column for $\neg p \land \neg q$.

(D) Creating a column for $(p \land q) \lor r$.

Question 6. Which of the following statements is NOT logically equivalent to $p \implies q$?

(A) $\neg p \lor q$

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

(C) $p \land \neg q$

(D) It is not the case that ($p$ is true and $q$ is false).

Question 7. De Morgan's Laws state that $\neg (p \land q) \equiv \neg p \lor \neg q$ and $\neg (p \lor q) \equiv \neg p \land \neg q$. Which statement is NOT a correct application or consequence of De Morgan's Laws?

(A) The negation of an "AND" statement is an "OR" statement of the negations.

(B) The negation of an "OR" statement is an "AND" statement of the negations.

(C) $\neg (\text{All are P}) \equiv \text{All are not P}$.

(D) $\neg (\text{Some are P}) \equiv \text{None are P}$.

Question 8. If a compound statement is a contingency, its truth table's final column will NOT:

(A) Contain only 'True' values.

(B) Contain only 'False' values.

(C) Be all the same truth value.

(D) Contain at least one 'True' and at least one 'False'.

Question 9. Which logical equivalence is NOT correct?

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

(B) $p \lor \text{False} \equiv p$

(C) $p \land \text{False} \equiv \text{False}$

(D) $p \lor \text{True} \equiv \text{False}$

Question 10. Which statement about logical equivalence is NOT true?

(A) If $P \equiv Q$, then $Q \equiv P$.

(B) If $P \equiv Q$ and $Q \equiv R$, then $P \equiv R$.

(C) $P \equiv Q$ means $P$ and $Q$ are the same statement.

(D) $P \equiv Q$ means $P \iff Q$ is a tautology.

Question 1. Which is NOT a method used for validating mathematical statements?

(A) Direct Proof

(B) Proof by Contradiction

(C) Proof by Example (for universal statements)

(D) Proof by Contrapositive

Question 2. In a direct proof of $p \implies q$, which statement is NOT an assumed starting point?

(A) $p$ is true.

(B) The given axioms and definitions are true.

(C) Previously proven theorems are true.

(D) $q$ is true.

Question 3. Which statement is NOT true about proof by contrapositive of $p \implies q$?

(A) You prove $\neg q \implies \neg p$.

(B) It relies on the logical equivalence of $p \implies q$ and $\neg q \implies \neg p$.

(C) You assume $\neg q$ and deduce $\neg p$.

(D) You assume $p$ and deduce $q$.

Question 4. In a proof by contradiction for a statement $P$, which is NOT a typical step?

(A) Assume $\neg P$ is true.

(B) Use logical steps to derive a contradiction.

(C) Conclude that $\neg P$ must be false.

(D) Conclude that $P$ must be false.

Question 5. An argument is valid if:

(A) The conclusion is true whenever all the premises are true.

(B) All the premises are true.

(C) The corresponding conditional (premises implies conclusion) is a tautology.

(D) It is impossible for all premises to be true and the conclusion false.

Question 6. Which of the following is NOT a valid argument form?

(A) Modus Ponens

(B) Modus Tollens

(C) Fallacy of the Converse

(D) Hypothetical Syllogism

Question 7. A mathematical proof does NOT aim to show that a statement is:

(A) True

(B) Necessarily true

(C) Probably true

(D) Logically derived from axioms and definitions

Question 8. To disprove a universally quantified statement, which method is NOT generally used?

(A) Finding a counterexample.

(B) Proving the negation of the statement.

(C) Showing it leads to a contradiction (of the original statement).

(D) Using a direct proof.

Question 9. An argument is sound if it is valid and:

(A) The conclusion is true.

(B) The premises are true.

(C) The conclusion is false.

(D) The premises are false.

Question 10. Which statement about proving $p \implies q$ is NOT correct?

(A) Direct proof starts by assuming $p$ and ends by showing $q$.

(B) Proof by contrapositive starts by assuming $\neg q$ and ends by showing $\neg p$.

(C) Proof by contradiction starts by assuming $p \land \neg q$ and derives a contradiction.

(D) To prove $p \implies q$, you must show that $p$ is true.