Show that the relation R in the set A of all 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.
person
brightness_auto
1 Answer
(i) The number of pages in a book remain the same.
⇒ Relation R is reflexive.
(ii) The book x has the same number of pages as the book y.
⇒ Book y has the same number of pages as the book x.
⇒ The relation R is symmetric.
(iii) Books x and y have the same number of pages.
⇒ Also books y and z have the same number of pages.
⇒ Books x and z also have the same number of pages.
⇒ R is transitive also.
Thus, R is an equivalence relation.
⇒ Relation R is reflexive.
(ii) The book x has the same number of pages as the book y.
⇒ Book y has the same number of pages as the book x.
⇒ The relation R is symmetric.
(iii) Books x and y have the same number of pages.
⇒ Also books y and z have the same number of pages.
⇒ Books x and z also have the same number of pages.
⇒ R is transitive also.
Thus, R is an equivalence relation.