Paper 2 CS · UNIT 01 · MOCK TEST

Mock 3 – Mathematical Logic (Part 1)

Unit 1: Discrete Structures and Optimisation · 21 questions · about 25 minutes. Press End Mock any time to see your score and the correct answers.

All 14 topic-wise mocks in this unit

0 / 21 answered 00:00

Q1. Which of the following have true value if universe of discourse consist of all integers:
A. ∃n (2n = 3n)
B. ∀n (n² ≥ n)
C. ∀n (2n > n)
D. ∃n (n² = 2)
E. ∀x∀y ∃z (xy = z)
Choose the correct answer from the options given below:

AB, C and E only BA, B and E only CA, B, C and D only DB, C, D and E only

Q2. Match the LIST-I with LIST-II. Match the equivalent propositions in List-I and List-II

LIST-I	LIST-II
A. P→(Q→P)	I. (P∨Q)∧¬(P∧Q)
B. P→(Q∨R)	II. (P∨R)→Q
C. (P→Q)∧(R→Q)	III. (P→Q)∨(P→R)
D. ¬(P⇔Q)	IV. ¬P→(P→Q)

Choose the correct answer from the options given below:

AA-IV, B-II, C-III, D-I BA-IV, B-III, C-II, D-I CA-I, B-II, C-III, D-IV DA-I, B-III, C-II, D-IV

Q3. Match the LIST-I with LIST-II.

LIST-I	LIST-II
A. Commutative law	I. P∧(Q∨R) = (P∧Q)∨(P∧R)
B. Distributive law	II. P∧(P∨Q) = P
C. Demorgan law	III. P∧Q = Q∧P
D. Adsorption law	IV. ¬(P∨Q) = ¬P∧¬Q

Choose the correct answer from the options given below:

AA-III, B-IV, C-II, D-I BA-II, B-I, C-IV, D-III CA-III, B-I, C-IV, D-II DA-IV, B-II, C-III, D-I

Q4. Match List I with List II

List I	List II
A. If the Indian team wins, then it is raining	I. Inverse
B. If the Indian team does not win, then it is not raining	II. Converse
C. If it is raining, then the Indian team wins	III. Contrapositive
D. If it is not raining, then the Indian team does not win	IV. Conditional

AA-II, B-I, C-IV, D-III BA-III, B-I, C-II, D-IV CA-IV, B-I, C-II, D-III DA-III, B-II, C-IV, D-I

Q5. Let P denote "She is intelligent" and let Q denote "She is happy." Given below are the statements:
(a) If she is intelligent, then she is unhappy.
(b) She is neither intelligent nor happy.
(c) It is necessary to be not intelligent in order to be happy.
(d) To be not intelligent is to be unhappy.
Which of the following is the correct propositional expression for the above statements?

A(a) P → ¬Q; (b) P ∧ ¬Q; (c) Q → ¬P; (d) P → ¬Q B(a) P → Q; (b) ¬P ∧ ¬Q; (c) ¬Q → P; (d) ¬P → ¬Q C(a) P → ¬Q; (b) ¬P ∧ ¬Q; (c) Q → ¬P; (d) ¬P → ¬Q D(a) P → ¬Q; (b) P ∧ ¬Q; (c) Q → P; (d) ¬P → ¬Q

Q6. Match List I with List II: match the logically equivalent propositions

List I	List II
A. p → q	I. (p ∧ q) ∨ (¬p ∧ ¬q)
B. ¬(p ∨ (¬p ∧ q))	II. ¬p ∨ q
C. p ↔ q	III. ¬(p ∨ q)
D. ¬(p ↔ q)	IV. ¬p ↔ q

AA-I, B-III, C-II, D-IV BA-I, B-II, C-III, D-IV CA-II, B-III, C-I, D-IV DA-II, B-III, C-IV, D-I

Q7. Let p be "It is a hot day" and q be "The temperature is 48°C". Write in simple sentences the meaning of ¬p ∧ ¬q.

AIt is a hot day or the temperature is 48°C BIt is a cold day or the temperature is 48°C CIt is neither a hot day nor is the temperature 48°C DIt is not a hot day

Q8. Consider the following, where p and q are two statements.
(A) ~(p ∧ q) ≡ ~p ∨ ~q
(B) ~(p ∨ q) ≡ ~p ∧ ~q
(C) p ∧ ~p ≡ T
(D) ~(p ⇒ q) ≡ p ∧ ~q
(E) p ∨ q ≡ ~p ∨ ~q
Which of these equivalences are correct?

A(A), (B) and (D) only B(A), (C) and (D) only C(C), (D) and (E) only D(A), (B) and (C) only

Q9. Let L(x, y) be the statement "x loves y", where the domain for both x and y consists of all people in the world. Use quantifiers to express "Joy is loved by everyone".

A∀x L(x, Joy) B∀y L(Joy, y) C∃y ∀x L(x, y) D∃x ¬L(Joy, x)

Q10. Translate ∀x ∃y (x < y) into English, considering the domain as real numbers for both variables.

AFor all real numbers x, there exists a real number y such that x is less than y BFor every real number y, there exists a real number x such that x is less than y CFor some real number x, there exists a real number y such that x is less than y DFor each and every real number x and y, x is less than y

Q11. If the universe of discourse is all real numbers, which of the following are true?
(A) ∃x ∀y (x + y = y)
(B) ∀x ∀y (((x ≥ 0) ∧ (y < 0)) → (x − y > 0))
(C) ∃x ∃y (((x ≤ 0) ∧ (y ≤ 0)) ∧ (x − y > 0))
(D) ∀x ∀y ((x ≠ 0) ∧ (y ≠ 0) ↔ (xy ≠ 0))
Choose the correct answer from the options given below:

A(A) and (B) Only B(A), (C) and (D) Only C(A), (B) and (D) Only D(A), (B), (C) and (D)

Q12. If the universe of discourse is the set of integers, which of the following are TRUE?
(A) ∀n ∃m (n² < m)
(B) ∃n ∀m (n < m²)
(C) ∃n ∀m (nm = m)
(D) ∃n ∃m (n² + m² = 6)
(E) ∃n ∃m (n + m = 4 ∧ n − m = 1)
Choose the correct answer from the options given below:

A(A), (B) and (C) Only B(B) and (C) Only C(C), (D) and (E) Only D(C) and (E) Only

Q13. Which of the following are tautologies?
(A) (P → (P → Q)) → (P → Q)
(B) ((P → Q) → Q) → (P ∨ Q)
(C) ((P ∨ ¬P) → Q) → ((P ∨ ¬P) → R)
(D) (Q → (P ∧ ¬P)) → (R → (P ∧ ¬P))
Choose the correct answer from the options given below:

A(A) Only B(B) Only C(A) and (B) Only D(C) and (D) Only

Q14. Let P(x) be the statement "x = x²". If the universe of discourse is the integers, which of the following have truth value TRUE?
(A) P(0)
(B) P(1)
(C) P(2)
(D) ∃x P(x)
(E) ∀x P(x)
Choose the correct answer from the options given below:

A(A), (B) and (E) Only B(A), (B) and (C) Only C(A), (B) and (D) Only D(B), (C) and (D) Only

Q15. Which of the following statements are true if the universe of discourse is the set of integers?
(A) ∀n (n² ≥ 0)
(B) ∃n (n² = 2)
(C) ∀n (n² ≥ n)
(D) ∃n (n² < 0)
Choose the correct answer from the options given below:

A(A) and (B) Only B(B) and (C) Only C(C) and (D) Only D(A) and (C) Only

Q16. Match List I with List II (logically equivalent expressions)

List I	List II
(a) P ∧ (P → Q)	(I) P ∨ Q
(b) ¬(P ∨ Q) → (P ∧ Q)	(II) (P ∧ ¬P) ∨ (P ∧ Q)
(c) P → Q	(III) (¬P) ∨ Q
(d) P ∨ (Q ∧ R)	(IV) (P ∧ P) ∨ (P ∧ Q) ∨ (P ∧ R) ∨ (Q ∧ R)

A(a)-(I), (b)-(II), (c)-(III), (d)-(IV) B(a)-(II), (b)-(I), (c)-(III), (d)-(IV) C(a)-(III), (b)-(I), (c)-(II), (d)-(IV) D(a)-(IV), (b)-(III), (c)-(II), (d)-(I)

Q17. Consider the universe of positive integers X = {n | 1 ≤ n ≤ 8}, proposition P = "n is an even integer", Q = "(3 ≤ n ≤ 7) ∧ (n ≠ 6)". The truth set of P ↔ Q is

A{1, 4} B{2, 6} C{3, 4, 5} D{1}

Q18. Which of the following are TRUE if the universe of discourse consists of all integers?
A. ∀n ∃m (n² < m)
B. ∃n ∀m (n < m²)
C. ∃n ∃m (n² + m² = 5)
D. ∃n ∃m (n² + m² = 6)
E. ∃n ∃m (m + n = 4 ∧ n − m = 1)
Choose the correct answer from the options given below:

AA, B, C Only BB, C, E Only CC, D, E Only DB, D Only

Q19. Which of the following are tautologies?
A. (p ∨ r) → p
B. p → (p ∧ r)
C. ~p → (p → r)
D. (p ∧ r) → (p → r)
E. ~(p → r) → p
Choose the correct answer from the options given below:

AA, B Only BA, B, C Only CB, C, D Only DC, D, E Only

Directions (Q91–95): A firm can produce two types of cloth, A and B. Three kinds of wool are required: red, green and blue. One unit length of type A cloth needs 2 yards of red and 3 yards of green wool. One unit length of type B cloth needs 3 yards of red wool, 2 yards of green wool and 2 yards of blue wool. The firm has a stock of only 8 yards of red wool, 10 yards of green wool and 15 yards of blue wool. The income from one unit length of type A and type B cloth is ₹30 and ₹50 respectively. Let x and y be the unit lengths of type A and type B cloth produced.

Q20. On the basis of the available material, the maximum total income is:

A₹133.3 B₹124 C₹100 D₹153.3

Q21. Which of the following is NOT logically equivalent to P → Q?

A~P ∨ Q B~Q → ~P C~(P ∧ ~Q) D~Q → P

Next: Mock 4 – Mathematical Logic (Part 2) →