Single Best Answer MCQs for Sub-Topics of Topic 11: Mathematical Reasoning

Question 1. Which of the following is considered a mathematical statement?

(A) Close the door!

(B) What is the time?

(C) $x + 5 = 10$

(D) The sum of two even integers is an even integer.

Question 2. A proposition is a sentence that:

(A) Is true or false, but not both.

(B) Is a question or a command.

(C) Contains a variable.

(D) Expresses an opinion.

Question 3. Consider the sentence: "This statement is false." Is this a valid mathematical statement?

(A) Yes, its truth value can be determined.

(B) No, it leads to a paradox.

(C) Yes, it is either true or false.

(D) No, it is a command.

Question 4. What is the truth value of the statement "Mumbai is the capital of India"?

(A) True

(B) False

(C) Cannot be determined

(D) Depends on the context

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

(A) $2 + 3 = 5$

(B) The square of any real number is non-negative.

(C) $x$ is an even number.

(D) $\pi > 3.14$

Question 6. Which of the following is a proposition?

(A) What a beautiful day!

(B) $y > 7$

(C) The sum of angles in a triangle is $180^\circ$.

(D) Please pass the salt.

Question 7. Identify the open sentence among the following:

(A) $7$ is a prime number.

(B) The Sun rises in the east.

(C) $x^2 = 25$

(D) $10 - 2 = 8$

Question 8. The truth value of the statement "Every square is a rectangle" is:

(A) True

(B) False

(C) Undetermined

(D) Depends on the specific square

Question 9. Which sentence cannot be assigned a definite truth value?

(A) Varanasi is in Uttar Pradesh.

(B) $10 + 5 = 16$

(C) Is it cold outside?

(D) The product of any two odd numbers is odd.

Question 10. A mathematical statement must be a declaration that is:

(A) True

(B) False

(C) Either true or false

(D) A question

Question 11. Consider the statement: "For any real number $x$, $x^2 \geq 0$". What is its truth value?

(A) True

(B) False

(C) Depends on $x$

(D) Cannot be determined

Question 12. Which of these sentences is a proposition?

(A) Go home now!

(B) How old are you?

(C) $5x + 2 > 12$

(D) The Earth is flat.

Question 13. Is the sentence "$x$ is a student in this class" a statement?

(A) Yes

(B) No, it's an open sentence.

(C) Yes, its truth value is obvious.

(D) Yes, it can be true or false.

Question 14. The truth value of the statement "$2$ is the only even prime number" is:

(A) True

(B) False

(C) Debatable

(D) Depends on the number system

Question 15. An opinion like "Red is the best colour" is generally not considered a mathematical statement because:

(A) It is false.

(B) It is true for some people.

(C) Its truth value is subjective/not universally agreed upon.

(D) It contains a colour.

Question 16. Which of these is a valid mathematical statement?

(A) $x+y > 10$

(B) What is your favourite number?

(C) For every integer $n$, $n+1$ is an integer.

(D) This is a good book.

Question 17. Is the sentence "He is a tall person" a statement?

(A) Yes, if we know who "He" is.

(B) No, "tall" is subjective.

(C) Yes, it can be true or false.

(D) No, it's a description.

Question 18. The truth value of the statement "The sum of interior angles of a quadrilateral is $360^\circ$" is:

(A) True

(B) False

(C) Sometimes true, sometimes false

(D) Undetermined

Question 19. An imperative sentence like "Do your homework" is not a statement because:

(A) It is false.

(B) It is true.

(C) It cannot be assigned a truth value (true or false).

(D) It is a command.

Question 20. Which of the following sentences is NOT a proposition?

(A) The product of $3$ and $4$ is $12$.

(B) Delhi is smaller than Mumbai.

(C) Could you help me with this problem?

(D) $\sqrt{2}$ is an irrational number.

Question 21. The truth value of the statement "$1$ is a prime number" is:

(A) True

(B) False

(C) Depends on definition

(D) Cannot be determined

Question 22. Is the sentence "$x^2 + 1 = 0$" a statement?

(A) Yes, it is a mathematical statement.

(B) No, it's an equation, not a statement.

(C) Yes, but its truth value depends on the domain of $x$.

(D) No, it's an open sentence because of the variable $x$.

Question 23. Which property defines a mathematical statement?

(A) It must contain numbers.

(B) It must be universally true.

(C) It must be either true or false.

(D) It must be a sentence.

Question 24. Consider the sentence "He is a doctor". For this to be a statement, we need to know:

(A) What "He" means.

(B) What a doctor is.

(C) Whether "He" likes his job.

(D) The truth value of the sentence.

Question 25. The statement "$2 \times 2 = 4$" has a truth value of:

(A) True

(B) False

(C) Undefined

(D) Both true and false

Question 1. If $p$ is the statement "It is raining", what is the negation of $p$, denoted by $\neg p$ or $\sim p$?

(A) It is not raining.

(B) It is sunny.

(C) Is it raining?

(D) Let it rain.

Question 2. Write the negation of the statement "All students are present".

(A) All students are absent.

(B) No students are present.

(C) Some students are not present.

(D) Some students are present.

Question 3. A compound statement is formed by connecting two or more simple statements using:

(A) Variables

(B) Logical connectives

(C) Numbers

(D) Truth values

Question 4. If $p$ is "The sky is blue" and $q$ is "Grass is green", which of the following represents a compound statement?

(A) The sky is blue.

(B) Is grass green?

(C) The sky is blue and grass is green.

(D) Grass is not green.

Question 5. What is the negation of the statement "$5 > 3$"?

(A) $5 < 3$

(B) $5 \leq 3$

(C) $5 = 3$

(D) $3 > 5$

Question 6. Let $p$ be the statement "The car is red". Which of the following is the correct symbolical representation of its negation?

(A) $p$

(B) $p \land q$

(C) $\neg p$

(D) $p \implies q$

Question 7. The negation of "No birds can swim" is:

(A) All birds can swim.

(B) Some birds can swim.

(C) Some birds cannot swim.

(D) No birds cannot swim.

Question 8. Which of the following is a compound statement?

(A) Goats eat grass.

(B) Is $5$ a prime number?

(C) If it rains, then the ground is wet.

(D) Mathematics is difficult.

Question 9. The negation of a statement reverses its:

(A) Meaning

(B) Truth value

(C) Structure

(D) Subject

Question 10. If a simple statement $p$ is false, what is the truth value of $\neg p$?

(A) True

(B) False

(C) Undetermined

(D) Both True and False

Question 11. The statement "It is cold and it is sunny" is a compound statement formed using which connective?

(A) Negation

(B) Conjunction

(C) Disjunction

(D) Conditional

Question 12. Write the negation of the statement "My friend is tall or my friend is thin".

(A) My friend is not tall or my friend is not thin.

(B) My friend is not tall and my friend is not thin.

(C) My friend is tall and my friend is thin.

(D) My friend is not tall or my friend is thin.

Question 13. Which symbol is commonly used for negation?

(A) $\land$

(B) $\lor$

(C) $\neg$

(D) $\implies$

Question 14. How is the negation of a simple statement formed?

(A) By adding "and" to it.

(B) By adding "not" or an equivalent phrase.

(C) By making it a question.

(D) By reversing the words.

Question 15. Let $p$ be "It is warm". The negation $\neg p$ is:

(A) It is cold.

(B) It is not warm.

(C) Is it warm?

(D) It was warm.

Question 16. Which of these is NOT a logical connective used to form compound statements?

(A) AND

(B) OR

(C) BUT

(D) NOT

Question 17. The statement "Some fruits are sweet" is the negation of:

(A) All fruits are sweet.

(B) No fruits are sweet.

(C) Some fruits are not sweet.

(D) All fruits are not sweet.

Question 18. If statement $p$ is true and statement $q$ is false, which of the following compound statements is true?

(A) $p \land q$

(B) $p \lor q$

(C) $\neg p \land q$

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

Question 19. The statement "It is not the case that it is raining" is equivalent to:

(A) It is raining.

(B) It is not raining.

(C) It might rain.

(D) It stopped raining.

Question 20. Which symbol represents the conjunction of two statements?

(A) $\lor$

(B) $\land$

(C) $\neg$

(D) $\iff$

Question 21. If $p$ is "I will go" and $q$ is "You will stay", the compound statement "I will go and you will stay" is symbolised as:

(A) $p \lor q$

(B) $p \land q$

(C) $\neg p \land q$

(D) $p \implies q$

Question 22. The statement "The integer is positive or the integer is negative" is a compound statement using which connective?

(A) AND

(B) NOT

(C) OR

(D) IF...THEN

Question 23. The negation of "The number is even" is:

(A) The number is odd.

(B) The number is not even.

(C) The number is odd or the number is not even.

(D) The number is not odd.

Question 24. Which of the following statements represents the negation of $p$?

(A) It is true that $p$.

(B) It is false that $p$.

(C) $p$ is possible.

(D) $p$ is necessary.

Question 25. Let $p$ be true and $q$ be true. What is the truth value of $\neg (p \land q)$?

(A) True

(B) False

(C) Undetermined

(D) Cannot be evaluated

Question 1. The symbol $\land$ represents which logical connective?

(A) OR

(B) NOT

(C) AND

(D) IF...THEN

Question 2. What is the truth value of $p \land q$ if $p$ is true and $q$ is false?

(A) True

(B) False

(C) Cannot be determined

(D) True or False depending on the statements

Question 3. The symbol $\lor$ represents which logical connective?

(A) AND

(B) NOT

(C) OR

(D) IF AND ONLY IF

Question 4. What is the truth value of $p \lor q$ if $p$ is false and $q$ is true (using inclusive OR)?

(A) True

(B) False

(C) Cannot be determined

(D) True and False

Question 5. What is the truth value of $\neg p$ if $p$ is true?

(A) True

(B) False

(C) Cannot be determined

(D) Both True and False

Question 6. How many rows are needed in the truth table for the compound statement $(p \land q) \lor \neg r$?

(A) 2

(B) 4

(C) 6

(D) 8

Question 7. Let $p$ be "I will eat rice" and $q$ be "I will eat roti". The statement "I will eat rice or roti" can be written as:

(A) $p \land q$

(B) $p \lor q$

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

(D) $p \implies q$

Question 8. The truth table for $p \land q$ has which output column?

(A) T F T F

(B) T T T F

(C) T F F F

(D) F F F T

Question 9. The truth table for $p \lor q$ (inclusive) has which output column?

(A) T F T F

(B) T T T F

(C) T F F F

(D) F F F T

Question 10. If statement $p$ is false, what is the truth value of $p \land q$ for any statement $q$?

(A) True

(B) False

(C) Depends on $q$

(D) Cannot be determined

Question 11. If statement $p$ is true, what is the truth value of $p \lor q$ for any statement $q$?

(A) True

(B) False

(C) Depends on $q$

(D) Cannot be determined

Question 12. Which truth table column represents the negation $\neg p$ if the column for $p$ is T F T F?

(A) T F T F

(B) F T F T

(C) T T F F

(D) F F T T

Question 13. What is the truth value of "The Sun is a star and the Moon is a planet"?

(A) True

(B) False

(C) Undetermined

(D) Depends on the definition of planet

Question 14. What is the truth value of "The Ganges is a river or Mount Everest is a mountain"?

(A) True

(B) False

(C) Undetermined

(D) Both are false

Question 15. Consider the statement "You can have tea or coffee" in a restaurant. This is typically interpreted as:

(A) Exclusive OR

(B) Inclusive OR

(C) Conjunction

(D) Negation

Question 16. If $p$ is true, $q$ is false. What is the truth value of $\neg p \lor q$?

(A) True

(B) False

(C) Cannot be determined

(D) Same as $p \lor q$

Question 17. How many possible truth assignments are there for a compound statement involving 3 simple propositions?

(A) 3

(B) 6

(C) 8

(D) 9

Question 18. If $p$ is false and $q$ is false, what is the truth value of $p \land q$?

(A) True

(B) False

(C) Depends on the statements

(D) Undetermined

Question 19. If $p$ is false and $q$ is false, what is the truth value of $p \lor q$?

(A) True

(B) False

(C) Depends on the statements

(D) Undetermined

Question 20. The statement "It is raining or it is not raining" is an example of a statement structure that will always be:

(A) True

(B) False

(C) Contingent

(D) Meaningless

Question 21. What is the truth value of $(\neg p) \land p$ for any truth value of $p$?

(A) True

(B) False

(C) Depends on $p$

(D) True if $p$ is true, False if $p$ is false

Question 22. How many columns are needed in a truth table for $\neg (p \lor \neg q)$?

(A) 2

(B) 3

(C) 4

(D) 5

Question 23. The exclusive OR (XOR), denoted by $\oplus$, is true when:

(A) Both statements are true.

(B) Both statements are false.

(C) Exactly one statement is true.

(D) At least one statement is true.

Question 24. If $p$ is true and $q$ is true, what is the truth value of $p \oplus q$ (exclusive OR)?

(A) True

(B) False

(C) Cannot be determined

(D) Same as $p \lor q$

Question 25. The connective "neither... nor..." between $p$ and $q$ is logically equivalent to:

(A) $p \land q$

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

(C) $p \lor q$

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

Question 1. The statement "$p \implies q$" is read as:

(A) p and q

(B) p or q

(C) If p, then q

(D) p if and only if q

Question 2. A conditional statement "$p \implies q$" is false only 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. The statement "$p \iff q$" is read as:

(A) If p, then q

(B) p and q

(C) p or q

(D) p if and only if q

Question 4. A biconditional statement "$p \iff q$" is true when:

(A) p and q have the same truth value.

(B) p and q have different truth values.

(C) p is true and q is false.

(D) p is false and q is true.

Question 5. In the conditional statement "If it rains (p), then the ground is wet (q)", $q$ is called the:

(A) Antecedent

(B) Consequent

(C) Implication

(D) Hypothesis

Question 6. If the statement "If I study hard (p), then I will pass (q)" is true, and I did not pass, what can be concluded about my studying?

(A) I must have studied hard.

(B) I must not have studied hard.

(C) I might have studied hard.

(D) No conclusion can be drawn.

Question 7. The statement "$p$ is sufficient for $q$" is equivalent to:

(A) $q \implies p$

(B) $p \implies q$

(C) $p \land q$

(D) $p \lor q$

Question 8. The statement "$q$ is necessary for $p$" is equivalent to:

(A) $q \implies p$

(B) $p \implies q$

(C) $p \land q$

(D) $p \lor q$

Question 9. If $p$ is false and $q$ is true, what is the truth value of $p \implies q$?

(A) True

(B) False

(C) Cannot be determined

(D) Same as $q \implies p$

Question 10. If $p$ is false and $q$ is false, what is the truth value of $p \implies q$?

(A) True

(B) False

(C) Cannot be determined

(D) Depends on the statements

Question 11. Which of the following represents the biconditional statement?

(A) $p \land q$

(B) $p \lor q$

(C) $p \implies q$

(D) $p \iff q$

Question 12. The statement "$p \iff q$" is logically equivalent to:

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

(B) $(p \lor q) \land (p \land q)$

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

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

Question 13. If $p$ is true and $q$ is false, what is the truth value of $p \iff q$?

(A) True

(B) False

(C) Cannot be determined

(D) Same as $q \iff p$

Question 14. The phrase "only if" introduces the:

(A) Antecedent

(B) Consequent

(C) Negation

(D) Conjunction

Question 15. The statement "I will go to the market if and only if it does not rain". Let $p$ be "I will go to the market" and $q$ be "It rains". The symbolic form is:

(A) $p \iff q$

(B) $q \iff p$

(C) $p \iff \neg q$

(D) $\neg q \iff \neg p$

Question 16. When is the statement "If the sky is green, then $2+2=5$" true?

(A) It is always true.

(B) It is always false.

(C) Only when the sky is green and $2+2=5$.

(D) Since the antecedent is false, the implication is true.

Question 17. Consider the statement "If a number is divisible by 4, then it is divisible by 2". Is this conditional statement true or false?

(A) True

(B) False

(C) Depends on the number

(D) Cannot be determined

Question 18. The statement "$p$ is a necessary condition for $q$" is equivalent to:

(A) $p \implies q$

(B) $q \implies p$

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

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

Question 19. When is the biconditional $p \iff q$ false?

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

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

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

(D) When $p$ and $q$ have the same truth values.

Question 20. Let $p$ be "$x=2$" and $q$ be "$x^2=4$". The statement "$x=2$ if and only if $x^2=4$" is:

(A) True

(B) False

(C) Depends on the domain of $x$

(D) Meaningless

Question 21. The contrapositive of $p \implies q$ is logically equivalent to:

(A) $p \implies q$

(B) $q \implies p$

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

(D) $p \land \neg q$

Question 22. The phrase "$p$ implies $q$" is equivalent to:

(A) $p$ is necessary for $q$.

(B) $q$ is necessary for $p$.

(C) $p$ if $q$.

(D) $q$ if $p$.

Question 23. If $p \implies q$ is true and $p$ is true, what can we conclude about $q$?

(A) $q$ must be true.

(B) $q$ must be false.

(C) $q$ can be true or false.

(D) No conclusion.

Question 24. If $p \implies q$ is true and $q$ is false, what can we conclude about $p$?

(A) $p$ must be true.

(B) $p$ must be false.

(C) $p$ can be true or false.

(D) No conclusion.

Question 25. The statement "$p \iff q$" means that $p$ is true under the same conditions that $q$ is true. This is expressed as:

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

(B) $p$ is a sufficient condition for $q$.

(C) $p$ is a necessary and sufficient condition for $q$.

(D) $p$ is neither necessary nor sufficient for $q$.

Question 1. Given the conditional statement "$p \implies q$", the converse is:

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

(B) $q \implies p$

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

(D) $p \iff q$

Question 2. Given the conditional statement "$p \implies q$", the contrapositive is:

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

(B) $q \implies p$

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

(D) $p \iff q$

Question 3. Which pair of statements is logically equivalent?

(A) A conditional statement and its converse.

(B) A conditional statement and its inverse.

(C) A conditional statement and its contrapositive.

(D) A conditional statement and its negation.

Question 4. Given the conditional statement "$p \implies q$", the inverse is:

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

(B) $q \implies p$

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

(D) $p \iff q$

Question 5. If a conditional statement is true, its converse is necessarily:

(A) True

(B) False

(C) Cannot be determined from the truth value of the original statement alone.

(D) Logically equivalent to the original statement.

Question 6. If a conditional statement is true, its contrapositive is necessarily:

(A) True

(B) False

(C) Cannot be determined from the truth value of the original statement alone.

(D) The negation of the original statement.

Question 7. The inverse of a conditional statement $p \implies q$ is logically equivalent to its:

(A) Converse

(B) Contrapositive

(C) Negation

(D) Original statement

Question 8. Write the converse of the statement "If it is a square, then it is a rectangle". Let $p$ be "it is a square" and $q$ be "it is a rectangle".

(A) If it is a rectangle, then it is a square.

(B) If it is not a square, then it is not a rectangle.

(C) If it is not a rectangle, then it is not a square.

(D) If it is a square, then it is not a rectangle.

Question 9. Write the contrapositive of the statement "If a number is even, then it is divisible by 2". Let $p$ be "a number is even" and $q$ be "it is divisible by 2".

(A) If a number is divisible by 2, then it is even.

(B) If a number is not even, then it is not divisible by 2.

(C) If a number is not divisible by 2, then it is not even.

(D) If a number is even, then it is not divisible by 2.

Question 10. Write the inverse of the statement "If a number is greater than 10, then it is greater than 5". Let $p$ be "a number is greater than 10" and $q$ be "it is greater than 5".

(A) If a number is greater than 5, then it is greater than 10.

(B) If a number is not greater than 10, then it is not greater than 5.

(C) If a number is not greater than 5, then it is not greater than 10.

(D) If a number is greater than 10, then it is not greater than 5.

Question 11. If the converse of $p \implies q$ is true, does $p \implies q$ necessarily have to be true?

(A) Yes

(B) No

(C) Only if $p$ and $q$ are true.

(D) Only if $p$ and $q$ are false.

Question 12. The logical equivalence between a conditional statement and its contrapositive is often used in which proof technique?

(A) Direct proof

(B) Proof by contradiction

(C) Proof by contrapositive

(D) Proof by induction

Question 13. If the statement "If a triangle is equilateral, then it is isosceles" is true, what is the truth value of its inverse?

(A) True

(B) False

(C) Depends on the triangle

(D) Cannot be determined

Question 14. The statement "If it is raining, then I will use an umbrella". Its converse is:

(A) If I will use an umbrella, then it is raining.

(B) If it is not raining, then I will not use an umbrella.

(C) If I will not use an umbrella, then it is not raining.

(D) It is raining and I will not use an umbrella.

Question 15. The statement "If a number is even, then it is divisible by 2". Its inverse is:

(A) If a number is divisible by 2, then it is even.

(B) If a number is not even, then it is not divisible by 2.

(C) If a number is not divisible by 2, then it is not even.

(D) A number is even and it is not divisible by 2.

Question 16. The statement "If a number is even, then it is divisible by 2". Its contrapositive is:

(A) If a number is divisible by 2, then it is even.

(B) If a number is not even, then it is not divisible by 2.

(C) If a number is not divisible by 2, then it is not even.

(D) A number is not even or it is divisible by 2.

Question 17. If the inverse of a statement is false, what is the truth value of the original conditional statement?

(A) True

(B) False

(C) Cannot be determined

(D) Same as the inverse

Question 18. If the converse of a statement is true, is the contrapositive of the original statement necessarily true?

(A) Yes

(B) No

(C) Only if the original is also true

(D) Only if the inverse is true

Question 19. The statement "$p \implies q$" is logically equivalent to:

(A) $\neg p \lor q$

(B) $\neg p \land q$

(C) $p \lor \neg q$

(D) $p \land \neg q$

Question 20. Which of the following is the negation of $p \implies q$?

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

(B) $q \implies p$

(C) $p \land \neg q$

(D) $\neg p \lor q$

Question 21. The inverse of the converse of $p \implies q$ is:

(A) $p \implies q$

(B) $q \implies p$

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

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

Question 22. If the statement "If you live in Delhi, then you live in India" is true, what is the truth value of its contrapositive?

(A) True

(B) False

(C) Depends on where you live

(D) Cannot be determined

Question 23. Consider the statement "If $n$ is an odd number, then $n^2$ is an odd number". Its converse is:

(A) If $n^2$ is an odd number, then $n$ is an odd number.

(B) If $n$ is not an odd number, then $n^2$ is not an odd number.

(C) If $n^2$ is not an odd number, then $n$ is not an odd number.

(D) If $n$ is an even number, then $n^2$ is an even number.

Question 24. Consider the statement "If $n$ is an odd number, then $n^2$ is an odd number". Its inverse is:

(A) If $n^2$ is an odd number, then $n$ is an odd number.

(B) If $n$ is not an odd number, then $n^2$ is not an odd number.

(C) If $n^2$ is not an odd number, then $n$ is not an odd number.

(D) If $n$ is an even number, then $n^2$ is an odd number.

Question 25. Consider the statement "If $n$ is an odd number, then $n^2$ is an odd number". Its contrapositive is:

(A) If $n^2$ is an odd number, then $n$ is an odd number.

(B) If $n$ is not an odd number, then $n^2$ is not an odd number.

(C) If $n^2$ is not an odd number, then $n$ is not an odd number.

(D) If $n$ is an even number, then $n^2$ is an even number.

Question 1. The symbol $\forall$ is called the:

(A) Existential quantifier

(B) Universal quantifier

(C) Negation symbol

(D) Conjunction symbol

Question 2. The statement "There exists a number $x$ such that $x^2 = 4$" uses which quantifier?

(A) Universal quantifier

(B) Existential quantifier

(C) Both universal and existential quantifiers

(D) No quantifier

Question 3. Write the negation of the statement "Some dogs can fly".

(A) Some dogs cannot fly.

(B) All dogs can fly.

(C) No dogs can fly.

(D) All dogs cannot fly.

Question 4. The statement "All birds can fly" can be represented symbolically as $\forall x, P(x)$ where $P(x)$ is:

(A) $x$ is a bird.

(B) $x$ can fly.

(C) $x$ is a bird and $x$ can fly.

(D) $x$ is a bird $\implies x$ can fly.

Question 5. The statement "There exists a prime number that is even" can be represented symbolically as $\exists x, P(x)$ where $P(x)$ is:

(A) $x$ is a prime number and $x$ is even.

(B) $x$ is a prime number or $x$ is even.

(C) $x$ is a prime number $\implies x$ is even.

(D) $x$ is a prime number if and only if $x$ is even.

Question 6. The negation of "Every student passed the exam" is:

(A) Every student did not pass the exam.

(B) No student passed the exam.

(C) Some students did not pass the exam.

(D) Some students passed the exam.

Question 7. The symbol $\exists$ is called the:

(A) Universal quantifier

(B) Existential quantifier

(C) Conditional symbol

(D) Biconditional symbol

Question 8. The statement "Some engineers are not rich" is the negation of:

(A) Some engineers are rich.

(B) All engineers are rich.

(C) No engineers are rich.

(D) All engineers are not rich.

Question 9. The statement "No animal can talk" is equivalent to:

(A) Some animals can talk.

(B) Some animals cannot talk.

(C) All animals cannot talk.

(D) There exists an animal that can talk.

Question 10. Which statement uses the universal quantifier?

(A) Some triangles are equilateral.

(B) There is a largest prime number.

(C) Every integer is a real number.

(D) There exists a solution to $x^2 = -1$ in complex numbers.

Question 11. Which statement uses the existential quantifier?

(A) All cats have tails.

(B) No circle is a square.

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

(D) There exists a triangle with angles $90^\circ, 45^\circ, 45^\circ$.

Question 12. The negation of the statement $\exists x, P(x)$ is logically equivalent to:

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

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

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

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

Question 13. The negation of the statement $\forall x, P(x)$ is logically equivalent to:

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

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

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

(D) $\exists x, P(x)$

Question 14. Write the negation of "There is a student who is taller than every teacher".

(A) There is a student who is shorter than every teacher.

(B) Every student is taller than some teacher.

(C) Every student is shorter than or equal to some teacher.

(D) For every student, there is a teacher who is taller than or equal to the student.

Question 15. The statement "Some equations have no solution" uses which quantifier?

(A) Universal

(B) Existential

(C) Both

(D) Neither

Question 16. The statement "Every multiple of 10 ends in 0" uses which quantifier implicitly?

(A) Universal

(B) Existential

(C) Neither

(D) Both

Question 17. The negation of "No integer is irrational" is:

(A) No integer is not irrational.

(B) Every integer is irrational.

(C) Some integer is irrational.

(D) Some integer is not irrational.

Question 18. The statement "Not all students like mathematics" is equivalent to:

(A) All students dislike mathematics.

(B) Some students like mathematics.

(C) Some students do not like mathematics.

(D) No student likes mathematics.

Question 19. The negation of "There is at least one solution" is:

(A) There is no solution.

(B) There is exactly one solution.

(C) There are multiple solutions.

(D) There are some solutions.

Question 20. The statement "Some numbers are even" can be written as $\exists x, P(x)$, where $P(x)$ is:

(A) $x$ is a number.

(B) $x$ is even.

(C) $x$ is a number and $x$ is even.

(D) If $x$ is a number, then $x$ is even.

Question 21. The statement "Every integer is positive" is:

(A) A universally quantified statement.

(B) An existentially quantified statement.

(C) A negation.

(D) A conjunction.

Question 22. The negation of "For every integer $n$, $n^2 \geq n$" is:

(A) For every integer $n$, $n^2 < n$.

(B) There exists an integer $n$ such that $n^2 \geq n$.

(C) There exists an integer $n$ such that $n^2 < n$.

(D) For every integer $n$, $n^2 \leq n$.

Question 23. The statement "There is no largest prime number" is an example involving:

(A) Universal quantifier

(B) Existential quantifier

(C) Negation and existential quantifier

(D) Negation and universal quantifier

Question 24. The statement "Some rectangles are squares" is true. Its negation is:

(A) Some rectangles are not squares.

(B) All rectangles are squares.

(C) No rectangle is a square.

(D) All squares are rectangles.

Question 25. The statement "Every dog is an animal" is equivalent to:

(A) Some dogs are animals.

(B) There exists a dog that is not an animal.

(C) If something is a dog, then it is an animal.

(D) If something is an animal, then it is a dog.

Question 1. A compound statement that is always true, regardless of the truth values of its simple statements, is called a:

(A) Contradiction

(B) Contingency

(C) Tautology

(D) Fallacy

Question 2. A compound statement that is always false, regardless of the truth values of its simple statements, is called a:

(A) Tautology

(B) Contingency

(C) Logical equivalence

(D) Contradiction (or Fallacy)

Question 3. Two statements $P$ and $Q$ are logically equivalent if:

(A) $P \land Q$ is a tautology.

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

(C) $P \lor Q$ is a tautology.

(D) $P \implies Q$ is a tautology.

Question 4. Which of the following statements is a tautology?

(A) $p \land \neg p$

(B) $p \lor \neg p$

(C) $p \implies \neg p$

(D) $p \iff \neg p$

Question 5. Which of the following statements is a contradiction?

(A) $p \land \neg p$

(B) $p \lor \neg p$

(C) $p \implies p$

(D) $p \lor p$

Question 6. A compound statement that is neither a tautology nor a contradiction is called a:

(A) Valid argument

(B) Logical equivalence

(C) Contingency

(D) Quantified statement

Question 7. To prove that $P$ is logically equivalent to $Q$ using a truth table, you must show that the truth value columns for $P$ and $Q$ are:

(A) Both all true.

(B) Both all false.

(C) Identical.

(D) Opposite.

Question 8. Which of the following statements is a contingency?

(A) $p \land \neg p$

(B) $p \lor \neg p$

(C) $p \implies q$

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

Question 9. The statement $\neg (\neg p)$ is logically equivalent to:

(A) $p$

(B) $\neg p$

(C) Tautology

(D) Contradiction

Question 10. De Morgan's law states that $\neg (p \land q)$ is logically equivalent to:

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

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

(C) $p \lor q$

(D) $p \land q$

Question 11. De Morgan's law states that $\neg (p \lor q)$ is logically equivalent to:

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

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

(C) $p \lor q$

(D) $p \land q$

Question 12. Which of the following is logically equivalent to $p \implies q$?

(A) $q \implies p$

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

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

(D) $p \land \neg q$

Question 13. How many rows are required in the truth table for a compound statement with $n$ distinct simple propositions?

(A) $n$

(B) $n^2$

(C) $2n$

(D) $2^n$

Question 14. The truth table column for a tautology consists of:

(A) All True values.

(B) All False values.

(C) A mix of True and False values.

(D) Only one True value.

Question 15. The truth table column for a contradiction consists of:

(A) All True values.

(B) All False values.

(C) A mix of True and False values.

(D) Only one False value.

Question 16. The statement $(p \land q) \implies p$ is:

(A) A tautology

(B) A contradiction

(C) A contingency

(D) Logically equivalent to $p$

Question 17. The statement $p \land (q \lor r)$ is logically equivalent to:

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

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

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

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

Question 18. Which of the following is the negation of $(p \lor q)$?

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

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

(C) $p \land q$

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

Question 19. Consider the truth table for $(p \land q) \implies (p \lor q)$. What kind of statement is it?

(A) Tautology

(B) Contradiction

(C) Contingency

(D) Cannot be determined

Question 20. The statement $p \lor (p \land q)$ is logically equivalent to:

(A) $p \land q$

(B) $p \lor q$

(C) $p$

(D) $q$

Question 21. The statement $p \land (p \lor q)$ is logically equivalent to:

(A) $p \land q$

(B) $p \lor q$

(C) $p$

(D) $q$

Question 22. Absorption law states that $p \lor (p \land q)$ is logically equivalent to $p$. This can be verified using a truth table with:

(A) 1 row

(B) 2 rows

(C) 4 rows

(D) 8 rows

Question 23. If the final column of a truth table for a compound statement contains at least one 'True' and at least one 'False', the statement is a:

(A) Tautology

(B) Contradiction

(C) Contingency

(D) Valid argument

Question 24. The statement $(\neg p \lor q) \lor (p \land \neg q)$ is:

(A) A tautology

(B) A contradiction

(C) A contingency

(D) Logically equivalent to $p \land q$

Question 25. Which of the following pairs of statements are NOT logically equivalent?

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

(B) $p \lor q$ and $\neg (\neg p \land \neg q)$

(C) $p \land q$ and $\neg (\neg p \lor \neg q)$

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

Question 1. Validating a mathematical statement means determining:

(A) If it is well-formed.

(B) If it is interesting.

(C) If it is true.

(D) If it can be negated.

Question 2. A proof by contradiction starts by assuming:

(A) The statement to be proven is true.

(B) The negation of the statement to be proven is true.

(C) The contrapositive of the statement is false.

(D) The converse of the statement is true.

Question 3. To prove a conditional statement $P \implies Q$ using the method of contrapositive, one would prove:

(A) $Q \implies P$

(B) $\neg P \implies \neg Q$

(C) $\neg Q \implies \neg P$

(D) $P \land \neg Q$ is a contradiction.

Question 4. A direct proof of $P \implies Q$ involves:

(A) Assuming $P$ is true and showing that $Q$ must be true.

(B) Assuming $\neg Q$ is true and showing that $\neg P$ must be true.

(C) Assuming $\neg (P \implies Q)$ is true and deriving a contradiction.

(D) Assuming $P$ and $Q$ are both true.

Question 5. An argument is considered valid if:

(A) The conclusion is true.

(B) The premises are true.

(C) The conclusion logically follows from the premises.

(D) Both premises and conclusion are true.

Question 6. Proof by contradiction is also known as:

(A) Direct proof

(B) Proof by contrapositive

(C) Indirect proof

(D) Proof by induction

Question 7. If an argument has true premises but a false conclusion, the argument is:

(A) Valid

(B) Invalid

(C) Sound

(D) Unsound

Question 8. The statement "$p$ implies $q$" means that the truth of $p$ guarantees the truth of $q$. This is the basis for which proof method?

(A) Proof by contradiction

(B) Proof by contrapositive

(C) Direct proof

(D) Proof by exhaustion

Question 9. To prove the statement "If $n^2$ is even, then $n$ is even" using contrapositive, you would prove:

(A) If $n$ is even, then $n^2$ is even.

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

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

(D) If $n^2$ is not even, then $n$ is not even.

Question 10. A mathematical proof is a step-by-step argument that shows a statement is:

(A) Probably true

(B) True in most cases

(C) Necessarily true

(D) Believable

Question 11. Modus Ponens is a valid argument form represented as:

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

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

(C) If $p$ then $q$. $\neg q$. Therefore $\neg p$.

(D) If $p$ then $q$. $\neg p$. Therefore $\neg q$.

Question 12. Modus Tollens is a valid argument form represented as:

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

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

(C) If $p$ then $q$. $\neg q$. Therefore $\neg p$.

(D) If $p$ then $q$. $\neg p$. Therefore $\neg q$.

Question 13. The Fallacy of the Converse has the form: If $p$ then $q$. $q$. Therefore $p$. Is this argument form valid?

(A) Yes

(B) No

(C) Only if $p$ and $q$ are true.

(D) Only if $p$ and $q$ are logically equivalent.

Question 14. The Fallacy of the Inverse has the form: If $p$ then $q$. $\neg p$. Therefore $\neg q$. Is this argument form valid?

(A) Yes

(B) No

(C) Only if the contrapositive is true.

(D) Only if the converse is false.

Question 15. What is the goal of a proof by contradiction for a statement $P$?

(A) To show that $P$ is true by assuming $P$ is true.

(B) To show that assuming $\neg P$ leads to a statement that is always false.

(C) To show that $P$ is equivalent to its contrapositive.

(D) To show that $P$ is a tautology.

Question 16. Which of the following is a valid argument?

(A) If it rains, I will stay home. I stayed home. Therefore, it rained.

(B) If it rains, I will stay home. It did not rain. Therefore, I did not stay home.

(C) If it rains, I will stay home. It rained. Therefore, I will stay home.

(D) If it rains, I will stay home. I will not stay home. Therefore, it rained.

Question 17. To show that the argument "Premise 1, Premise 2, ..., Premise n. Therefore, Conclusion" is valid using truth tables, you need to show that:

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

(B) All premises and the conclusion are true.

(C) The disjunction of the premises implies the conclusion.

(D) The conjunction of the premises is a tautology.

Question 18. A sound argument is a valid argument where:

(A) The conclusion is false.

(B) The premises are all true.

(C) The conclusion logically follows.

(D) It is a tautology.

Question 19. Consider the statement: "The sum of two odd integers is an even integer". Which proof method would typically start by letting the two odd integers be $2k+1$ and $2m+1$ for integers $k$ and $m$?

(A) Proof by contradiction

(B) Proof by contrapositive

(C) Direct proof

(D) Proof by cases

Question 20. A proof by contrapositive of $P \implies Q$ relies on the logical equivalence of $P \implies Q$ and:

(A) $Q \implies P$

(B) $\neg P \implies \neg Q$

(C) $\neg Q \implies \neg P$

(D) $P \land \neg Q$

Question 21. To prove that $\sqrt{2}$ is irrational using contradiction, you would start by assuming:

(A) $\sqrt{2}$ is rational.

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

(C) $\sqrt{2} = p/q$ where $p, q$ are integers, $q \neq 0$, and $\gcd(p,q)=1$.

(D) $\sqrt{2}$ is an integer.

Question 22. Which of the following argument forms is invalid?

(A) Modus Ponens

(B) Modus Tollens

(C) Fallacy of the Converse

(D) Disjunctive Syllogism (p or q, not p, therefore q)

Question 23. In a proof by contradiction of statement $P$, if the assumption $\neg P$ leads to a statement $R$ which is known to be true, does this prove $P$?

(A) Yes

(B) No

(C) Only if $R$ is a tautology.

(D) Only if $R$ is a contradiction.

Question 24. The purpose of checking the validity of an argument is to determine if:

(A) The premises are true.

(B) The conclusion is true.

(C) The reasoning from premises to conclusion is logically correct.

(D) The argument is convincing.

Question 25. If you are using a direct proof to show $P \implies Q$, you must begin by:

(A) Assuming $Q$ is true.

(B) Assuming $\neg P$ is true.

(C) Assuming $P$ is true.

(D) Assuming $\neg Q$ is true.