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 same number of pages} is an equivalence relation.

Answer 1

$\text{\bf{a relation will be equivalence relation}}\\\text{\bf{when the relation is :}}\\\text{\bf{reflexive}}\\\text{\bf{symmetric}}\\\text{\bf{transitive}}$

now, 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) ∈ R as x and x have same number of pages.
Let (x,x) ∈ R
⇒ x and y have the same number of pages
⇒ y and x have the same number of pages.
⇒ (y,x) ∈ R
Therefore, R is symmetric.
Now, let (x,y) ϵ 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
Therefore, R is transitive.
Therefore, R is an equivalence relation.