Sherlin and Danju are playing Ludo at home during Covid-19. While rolling the dice, Sherlin’s sister Raji observed and noted the possible outcomes of the throw every time belongs to set {1, 2, 3, 4, 5, 6}. Let A be the set of players while B be the set of all possible outcomes.
Sherlin and Danju are Playing Ludo - Teachoo.jpg



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

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Question 1

Let 𝑅 ∶ 𝐵 → 𝐵 be defined by R = {(𝑥, 𝑦): 𝑦 𝑖𝑠 𝑑𝑖𝑣𝑖𝑠𝑖𝑏𝑙𝑒 𝑏𝑦 𝑥 } is
(a) Reflexive and transitive but not symmetric
(b) Reflexive and symmetric and not transitive
(c) Not reflexive but symmetric and transitive
(d) Equivalence

 

Note : This question is same Ex 1.1, 1 (iii) - Chapter 1 Class 12 - Relations and Functions

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Question 2 – Concept (Number of functions)

Raji wants to know the number of functions from A to B. How many number of functions are possible?
(a) 6 2   (b) 2 6   (c) 6!   (d) 2 12

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Question 2

Raji wants to know the number of functions from A to B. How many number of functions are possible?
(a) 6 2   (b) 2 6   (c) 6!   (d) 2 12

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Question 3

Let R be a relation on B defined by R = {(1, 2), (2, 2), (1, 3), (3, 4), (3, 1), (4, 3), (5, 5)}. Then R is
(a) Symmetric
(b) Reflexive
(c) Transitive
(d) None of these three

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Question 4

Raji wants to know the number of relations possible from A to B. How many numbers of relations are possible?
(a) 6 2   (b) 2 6   (c) 6!   (d) 2 12

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Question 5

Let 𝑅: 𝐵 → 𝐵 be defined by R = {(1, 1), (1, 2), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6)}, then R is
(a) Symmetric
(b) Reflexive and Transitive
(c) Transitive and symmetric
(d) Equivalence

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Transcript

Question Sherlin and Danju are playing Ludo at home during Covid-19. While rolling the dice, Sherlin’s sister Raji observed and noted the possible outcomes of the throw every time belongs to set {1, 2, 3, 4, 5, 6}. Let A be the set of players while B be the set of all possible outcomes. A = {S, D}, B = {1, 2, 3, 4, 5, 6} Question 1 Let 𝑅 ∶ 𝐵 → 𝐵 be defined by R = {(𝑥, 𝑦): 𝑦 𝑖𝑠 𝑑𝑖𝑣𝑖𝑠𝑖𝑏𝑙𝑒 𝑏𝑦 𝑥 } is (a) Reflexive and transitive but not symmetric (b) Reflexive and symmetric and not transitive (c) Not reflexive but symmetric and transitive (d) Equivalence B = {1, 2, 3, 4, 5, 6} Given R = {(𝑥, 𝑦): 𝑦 𝑖𝑠 𝑑𝑖𝑣𝑖𝑠𝑖𝑏𝑙𝑒 𝑏𝑦 𝑥 } Check Reflexive Since x is divisible by x ∴ (x, x) ∈ R ∴ R is reflexive Check symmetric To check whether symmetric or not, If (x, y) ∈ R, then (y, x) ∈ R Here (2, 4) ∈ R , as 4 is divisible by 2 but (4, 2) ∉ R as 2 is not divisible by 4 ∴ R is not symmetric Check transitive If y is divisible by x & z is divisible by y, then z is divisible by x ∴ If (x, y) ∈ R & (y, z) ∈ R , then (x, z) ∈ R ∴ R is transitive Thus, R is reflexive and transitive but not symmetric So, the correct answer is (a) Question 2 – Concept (Number of functions) Raji wants to know the number of functions from A to B. How many number of functions are possible? (a) 62 (b) 26 (c) 6! (d) 212 Let A have 2 elements and B have 3 elements Thus, Number of functions = 3 × 3 = 32 = 9 Question 2 Raji wants to know the number of functions from A to B. How many number of functions are possible? (a) 62 (b) 26 (c) 6! (d) 212 Given A = {S, D}, B = {1, 2, 3, 4, 5, 6} So, A has 2 elements, B has 6 elements Numbers of functions from A to B = 62 Question 3 Let R be a relation on B defined by R = {(1, 2), (2, 2), (1, 3), (3, 4), (3, 1), (4, 3), (5, 5)}. Then R is (a) Symmetric (b) Reflexive (c) Transitive (d) None of these three Given R = {(1, 2), (2, 2), (1, 3), (3, 4), (3, 1), (4, 3), (5, 5)}. Check Reflexive Here, (1, 1) ∉ R ∴ R is not reflexive Check symmetric To check whether symmetric or not, If (x, y) ∈ R, then (y, x) ∈ R Here (1, 2) ∈ R , but (2, 1) ∉ R ∴ R is not symmetric Check transitive To check whether transitive or not, If (x, y) ∈ R & (y, z) ∈ R , then (x, z) ∈ R Here, (1, 3) ∈ R & (3, 4) ∈ R , but then (1, 4) ∉ R ∴ R is not transitive Thus, R is not reflexive, not symmetric and not transitive So, the correct answer is (d) Question 4 Raji wants to know the number of relations possible from A to B. How many numbers of relations are possible? (a) 62 (b) 26 (c) 6! (d) 212 Given A = {S, D}, B = {1, 2, 3, 4, 5, 6} Numbers of Relation from A to B = 2Numbers of elements of A × Number of elements of B = 22 × 6 = 212 So, the correct answer is (d) Question 5 Let 𝑅: 𝐵 → 𝐵 be defined by R = {(1, 1), (1, 2), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6)}, then R is (a) Symmetric (b) Reflexive and Transitive (c) Transitive and symmetric (d) Equivalence Given R = {(1, 1), (1, 2), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6)} Check Reflexive Here, (1, 1) ∈ R for all B = {1, 2, 3, 4, 5, 6} ∴ R is reflexive Check symmetric To check whether symmetric or not, If (x, y) ∈ R, then (y, x) ∈ R Here (1, 2) ∈ R , but (2, 1) ∉ R ∴ R is not symmetric Check transitive To check whether transitive or not, If (x, y) ∈ R & (y, z) ∈ R , then (x, z) ∈ R Since R = {(1, 1), (1, 2), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6)} For all values of x, y, z → (x, y) ∈ R & (y, z) ∈ R , then (x, z) ∈ R ∴ R is transitive Thus, R is reflexive and transitive So, the correct answer is (b)

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Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.