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Question
Let R be the relation defined on 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}.
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Solution
We observe the following properties of R.
Reflexivity :
Let a be an arbitrary element of R. Then,
a ∈ R
⇒ (a, a) ∈ R for all a ∈ A
So, R is reflexive on A.
Symmetry : Let (a, b) ∈ R
⇒ Both a and b are either even or odd.
⇒ Both b and a are either even or odd.
⇒ (b, a) ∈ R for all a, b ∈ A
So, R is symmetric on A.
Transitivity : Let (a, b) and (b, c) ∈ R
⇒ Both a and b are either even or odd and both b and c are either even or odd.
⇒ a, b and c are either even or odd.
⇒ a and c both are either even or odd.
⇒ (a, c) ∈ R for all a, c ∈ A
So, R is transitive on A.
Thus, R is an equivalence relation on A.
We observe that all the elements of the subset {1, 3, 5, 7} are odd. Thus, they are related to each other .
This is because the relation R on A is an equivalence relation.
Similarly, the elements of the subset {2, 4, 6} are even. Thus, they are related to each other because every element is even.
Hence proved .
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