Discrete Structures and Optimization – Previous Year Questions
272 questions from past UGC NET Paper 2 – Computer Science papers, each with the correct answer. Open any question for its explanation.
- Which of the following statements are true about sets? A. 0 ∈ ∅ B. ∅ ∈ {0} C. ∅ ∈ {∅} D. {∅} ∈ {∅} E. {∅} ⊂ {∅, {∅}} Choose the correct answer from the options given below:
- Consider the following relations on {1, 2, 3, 4}. Which of the following relations are reflexive? (A) R₁ = {(1,1),(1,2),(2,1),(2,2),(3,4),(4,1),(4,4)} (B) R₂ = {(1,1),(1,2),(2,1)} (C) R₃ = {(1,1),(1,2),(1,4),(2,1),(2,2)…
- 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.
- A graph G with n ≥ 3 vertices is a Hamiltonian graph if the degree of each vertex is greater than or equal to
- A coin is tossed successively three times. For the event E of getting exactly one head or exactly two heads, with sample space S (n denotes the number of outcomes): (A) n(S) = 8 and n(E) = 4 (B) n(E) = 6 and n(S) = 8 (C…
- 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?
- 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".
- Match List I (Queries) with List II (Probability)List I List II (A) A bag contains 6 white and 4 red balls. Two balls are drawn at random. What is the chance they will be the same colour? (I) 3/68 (B) In a pack of 52 ca…
- Let G(V, E) be an undirected graph with l edges. Then the sum of the degrees of all vertices is equal to
- 40 software professionals were interviewed for a job. 25 knew PYTHON, 20 knew JAVA and 7 knew neither language. How many knew both languages?
- Translate ∀x ∃y (x < y) into English, considering the domain as real numbers for both variables.
- What is the probability that a positive integer selected at random from the set of positive integers not exceeding 100 is divisible by either 2 or 5?
- 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) ∧ (…
- 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…
- If N² = N × N, where N is the set of natural numbers, and R is a relation on N² such that ⟨x, y⟩ R ⟨u, v⟩ ⇔ xv = yu, which of the following are TRUE? (A) Reflexive (B) Symmetric (C) Transitive (D) Asymmetric Choose the…
- 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 be…
- 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 optio…
- 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:
- Which of the following graphs (on vertices a–f) are trees? (A) edges: ac, bc, cd, cf, ef (B) edges: ac, af, df, ef, be (C) edges: ab, ac, ad, be, cf, de (D) edges: af, be, bd, ce Choose the correct answer from the optio…
- Match List I with List IIList I List II (A) φ ∩ {φ} (I) φ (B) {φ} ∩ {φ} (II) {φ} (C) {φ, {φ}} − φ (III) {{φ}} (D) φ ∪ {{φ}} (IV) {φ, {φ}}
- 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 ∧…
- Arrange the following in ascending order: (a) Remainder of 4916 when divided by 17 (b) Remainder of 2446 when divided by 9 (c) Remainder of 15517 when divided by 17 (d) Last digit of the number 745 Choose the correct an…
- Statement (I): If H is a non-empty finite subset of a group G and ab ∈ H for all a, b ∈ H, then H is also a group. Statement (II): There is no homomorphism from (Z, +) to (Q, +), where Z is the set of integers and Q the…
- The cost function of the total food is:
- Which of the following are the constraints when formulating the LPP?
- What are the quantities (in grams) of food X and Y respectively when the cost of food is minimum?
- The dual of the formulated LPP is:
- The minimum cost of food is:
- 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
- Let (Z, +) denote the group of all integers under addition. Then the number of automorphisms of (Z, +) is