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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}.

RELATIONS AND FUNCTION

Class 12, NCERT Chapter 1,  Example6 

Solution

Given any element a in A, both a and a must be either odd or even, so that (a, a) ∈ R.

 Further, (a, b) ∈ R ⇒ both a and b must be either odd or even ⇒ (b, a) ∈ R.

 Similarly, (a, b) ∈ R and (b, c) ∈ R ⇒ all elements a, b, c, must be either even or odd simultaneously ⇒ (a, c) ∈ R.

 Hence, R is an equivalence relation.

 Further, all the elements of {1, 3, 5, 7} are related to each other, as all the elements of this subset are odd. 

Similarly, all the elements of the subset {2, 4, 6} are related to each other, as all of them, are even.

 Also, no element of the subset {1, 3, 5, 7} can be related to any element of {2, 4, 6}, as elements of {1, 3, 5, 7} are odd, while elements of {2, 4, 6} are even.

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