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Question
Show that the relation R in the set A of all the books in a library of a college, given by R = {(x, y) : x and y have the same number of pages} is an equivalence relation.
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Solution
Set A is the set of all books in the library of a college.
(i) Reflexive:
R = {(x, y) : x and y have the same number of pages}
Since (x, x) ∈ R as x and x have the same number of pages.
∴ R is reflexive.
(ii) Symmetric:
Let (x, y) ∈ R
⇒ x and y have the same number of pages.
⇒ y and x have the same number of pages.
⇒ (y, x) ∈ R
∴ R is symmetric.
(iii) Transitive:
Now, let (x, y) ∈ R and (y, z) ∈ R.
⇒ x and y have the same number of pages and y and z have the same number of pages.
⇒ x and z have the same number of pages.
⇒ (x, z) ∈ R
∴ R is transitive.
Hence, R is an equivalence relation.
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