Misc 9 - Chapter 2 Class 11 Relations and Functions

Transcript

Misc 9 - Introduction 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 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 Since both conditions are not always true. ∴ (a, a) ∉ R Hence, the given statement is False Misc 9 - Introduction 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 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 Since, both conditions are not true. ∴ (b, a) ∉ R Hence, the given statement is False Misc 9 - Introduction 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. 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} Given (a, b) ∈ R , i.e. (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 Given (b, c) ∈ R , i.e. (b, c) is in relation R . So, the following conditions are true b, c ∈ N i.e. both b & c are natural numbers b = c2 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