Question

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 equivalencerelation. Further, show that all the elements of the subset {1, 3, 5, 7} are related to eachother and all the elements of the subset {2, 4, 6} are related to each other, but noelement of the subset {1, 3, 5, 7} is related to any element of the subset {2, 4, 6}.​

Answer 1

$\huge\star\mathfrak\pink{{Answer:-}}$

Solution Given any element a in A, both a and a must be either odd or even, sothat (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 beeither 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 elementsof this subset are odd. Similarly, all the elements of the subset {2, 4, 6} are related toeach 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 elementsof {2, 4, 6} are even.