Unit-II Worksheet PDF

IILM University, Greater Noida Unit-II Short Answer Questions 1. If 𝑅 = {(𝑥, 𝑦): 𝑥 + 2𝑦 = 8} is a relation on N, then write the range of 𝑅. 2. If 𝑃 = {1,3}, 𝑄 = {2,3,5} find the number of relations from 𝑃 to 𝑄. 3. Let 𝑅 be a relation on the set of na...

IILM University, Greater Noida Unit-II Short Answer Questions 1. If 𝑅 = {(𝑥, 𝑦): 𝑥 + 2𝑦 = 8} is a relation on N, then write the range of 𝑅. 2. If 𝑃 = {1,3}, 𝑄 = {2,3,5} find the number of relations from 𝑃 to 𝑄. 3. Let 𝑅 be a relation on the set of natural numbers 𝑁, as 𝑅 = {(𝑥, 𝑦) ∶ 𝑥, 𝑦  𝑁, 3𝑥 + 𝑦 = 19}. Find the domain and range of 𝑅. Verify whether R is reflexive. 4. Define reflexive, symmetric, antisymmetric and transitive relation. 5. What is partial order relation and how it is differ from equivalence relation? 6. Define closure relation and type of closure relation. 7. Check the below relations are reflexive, symmetric, antisymmetric and transitive. (i) 𝑅1 = {(1,1), (2,2), (3,3), (4,4), (1,2), (2,1)} (ii) 𝑅2 = {(1,1), (2,2), (3,3), (4,4)} (iii) 𝑅3 = {(1,1), (2,2), (3,3), (4,4), (1,2), (2,1), (1,3), (3,1), (4,1), (1,4)} Long Answer Questions 1. Let 𝑃 = {𝑎, 𝑏 , 𝑐, 𝑑, 𝑒, 𝑓} and 𝑄 = { 𝑥 , 𝑦, 𝑧, 𝑡, 𝑠}. Let 𝑅 be a relation from 𝑃 to 𝑄 defined by 𝑅 = { (𝑎, 𝑥 ), (𝑎, 𝑦), (𝑏, 𝑧), (𝑏, 𝑡), (𝑐, 𝑡), (𝑐, 𝑧), (𝑒, 𝑥 ), (𝑒, 𝑦)}. Find the domain ,range and inverse of 𝑅. 2. Let 𝑅 be a relation on the set of natural numbers 𝑁, as 𝑅 = {(𝑥, 𝑦) ∶ 𝑥, 𝑦  𝑁, 3𝑥 + 𝑦 = 19}. Find the domain and range of 𝑅. Verify whether R is reflexive. 3. The following relation on 𝐴 = {1, 2, 3, 4}. Determine whether the following: (a) 𝑅 = {(1, 3), (3,1), (1, 1), (1, 2), (3, 3), (4, 4)} poset (b) 𝑅 = 𝐴 × 𝐴 is an equivalence relation or not. 4. Let R be a binary relation defined as 𝑅 = { (𝑎, 𝑏) ∈ 𝑅2 ∶ (𝑎, 𝑏) ≤ 3} determine whether 𝑅 is reflexive, symmetric, antisymmetric and transitive and how many distinct binary relation are there on finite set. 5. Let 𝐴 = { 1,2,3,4,5,6 } and let R be the relation defined by x divides y written as 𝑥/𝑦 : (a) Write 𝑅 as a set of ordered pairs. (b) Find 𝑅−1 6. Show that the relation R in the set {1, 2, 3} given by 𝑅 = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 3)} is reflexive but neither symmetric nor transitive. 1