# 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 same number of pages} is an equivalence relation.

It is given that the set A of all the books in a library of a college. Then,

R = {(x, y): x and y have same number of pages}

Now, R is reflexive

Since, (x,x) ϵ R as x and x have same number of pages.

Let (x,x) ϵ R

x and y have the same number of pages

y and x have the same number of pages.

(y,x) ϵ R

Therefore, R is symmetric.

Now, let (x,y) ϵ R and (y,z) ϵ 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) ϵ R

Therefore, R is transitive.

Therefore, R is an equivalence relation.

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