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Learn in your speed, with individual attention - Teachoo Maths 1-on-1 Class


Transcript

Example 6 Let R be the relation defined in the set A = {1, 2, 3, 4, 5, 6, 7} by R = {(a, b) : both a and b are either odd or even}. Show that R is an equivalence relation. Further, show that all the elements of the subset {1, 3, 5, 7} are related to each other and all the elements of the subset {2, 4, 6} are related to each other, but no element of the subset {1, 3, 5, 7} is related to any element of the subset {2, 4, 6}. R = {(a, b) : both a and b are either odd or even} Check reflexive Since, a & a are the same numbers both a and a must be either odd or even, ∴ (a, a) ∈ R, So, R is reflexive Check symmetric If both a & b are either odd or even then, both b & a are either odd or even So, if (a, b) ∈ R , then (b, a)∈ R So, R is symmetric Check transitive If both a & b are either odd or even and both b & c are either odd or even , then a, b, c are either odd or even So, both a & c are either odd or even So, if (a, b) ∈ R and (b, c) ∈ R , then (a, c) ∈ R. So, R is transitive Since R is reflexive, symmetric and transitive Hence, R is an equivalence relation. Further, show that all the elements of the subset {1, 3, 5, 7} are related to each other and all the elements of the subset {2, 4, 6} are related to each other, but no element of the subset {1, 3, 5, 7} is related to any element of the subset {2, 4, 6}. R = {(a, b) : both a and b are either odd or even} In {1, 3, 5, 7}, All elements are odd, Hence, element of {1, 3, 5, 7 } are related to each other In {2, 4, 6}, All elements are even, Hence, element of {2, 4, 6} are related to each other In {1, 3, 5, 7} & {2, 4, 6}, Elements of {1, 3, 5, 7} are odd Elements of {2, 4, 6} are even One element from {1, 3, 5, 7} is odd and one element from {2, 4, 6} is even Hence, both elements cannot be either odd or even Hence, {1, 3, 5, 7} & {2, 4, 6} are not related to each other

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Davneet Singh

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.