Show that the relation R in the set A of all the books in a library of

1.	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» (x, x) ∈ R as x and x have same number of pages. ∀ x ∈ A, hence Reflexive (x, y) ∈ R ⇒ x and y have same number of pages ⇒ y and x have same number of pages ⇒ 9 (y, x) ∈ R hence Symmetric (x, y) ∈ R and (y, z) ∈ R ⇒ x and y and y and z have same number of pages ⇒ x and z have same number of pages ⇒ 9 (x, z) ∈ R, hence Transitive. Since R is reflexive, symmetric and transitive, R is an equivalence Relation.

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»

(x, x) ∈ R as x and x have same number of pages. ∀ x ∈ A, hence

Reflexive

(x, y) ∈ R ⇒ x and y have same number of pages

⇒ y and x have same number of pages

⇒ 9 (y, x) ∈ R hence Symmetric (x, y) ∈ R and (y, z) ∈ R

⇒ x and y and y and z have same number of pages

⇒ x and z have same number of pages

⇒ 9 (x, z) ∈ R, hence Transitive.

Since R is reflexive, symmetric and transitive, R is an equivalence Relation.