Last updated at Dec. 8, 2016 by Teachoo

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Misc 9 Let R be a relation from N to N defined by R = {(a, b): a, b ∈ N and a = b2}. Are the following true? (a, a) ∈ R, for all a ∈ N Introduction Is a = a2 Checking for different values of a 1 = 12 = 1 2 ≠ 22 3 ≠ 32 Hence, a = a2 is not always true Misc 9 Let R be a relation from N to N defined by R = {(a, b): a, b ∈ N and a = b2}. Are the following true? (a, a) ∈ R, for all a ∈ N Given R = {(a, b): a, b ∈ N and a = b2} Hence we can say that (a, b) is in relation R if a, b ∈ N i.e. both a & b are natural numbers a = b2 We need to check if both these conditions are true for (a,a) 1. a, a ∈ N , i.e. a is a natural number 2. a = a2 is not always true So, both conditions are not true. ∴ (a,a) ∉ R Hence the given statement is false Misc 9 Let R be a relation from N to N defined by R = {(a, b): a, b ∈ N and a = b2}. Are the following true? (ii) (a, b) ∈ R, implies (b, a) ∈ R Introduction If a = b2, then b = a2 ? Let b = 2, a = b2 = 22 = 4 but 2 ≠ 42 i.e. b ≠ a2 So, If a = b2, then b = a2 is not always true Misc 9 Let R be a relation from N to N defined by R = {(a, b): a, b ∈ N and a = b2}. Are the following true? (ii) (a, b) ∈ R, implies (b, a) ∈ R Given R = {(a, b): a, b ∈ N and a = b2} Given (a, b) is in relation R. So, the following conditions are true a, b ∈ N i.e. both a & b are natural numbers a = b2 We need to check if both these conditions are true for (b,a) 1. b, a ∈ N , i.e. b, a is a natural number 2. b = a2 is not always true So, both conditions are not true. ∴ (b,a) ∉ R Hence the given statement is false Misc 9 Let R be a relation from N to N defined by R = {(a, b): a, b ∈ N and a = b2}. Are the following true? (iii) (a, b) ∈ R, (b, c) ∈ R implies (a, c) ∈ R. Introduction If a = b2, & b = c2 , then a = c2? Let b = 4, a = b2 = 42 = 16 & 4 = c2 i.e. c = 2 But 16 ≠ 22 i.e. a ≠ c2 So, If a = b2, & b = c2 , then a = c2 is not always true Misc 9 Let R be a relation from N to N defined by R = {(a, b): a, b ∈ N and a = b2}. Are the following true? (iii) (a, b) ∈ R, (b, c) ∈ R implies (a, c) ∈ R. Given R = {(a, b): a, b ∈ N and a = b2} We need to prove both these conditions for (a, c) 1. Given a, b & b, c ∈ N, hence a, c ∈ N 2. If a = b2, & b = c2 , then a = c2 is not always true Since both the conditions are not true Hence, (a,c) ∉ R So, the given statement is false

About the Author

Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 7 years. He provides courses for Mathematics and Science from Class 6 to 12. You can learn personally from here https://www.teachoo.com/premium/maths-and-science-classes/.