Question

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.

Answer 1

Hint: Any relation can be classified as reflexive, symmetric and transitive. If aRa exists in the relation, then it is said to be reflexive. If aRb and bRa both exist in the relation, then it is said to be symmetric. If aRb and bRc exist implies that aRc also exists, the relation is transitive.

Complete step-by-step answer:
It is given that R = {(x, y): x and y have the same number of pages}. We have to show that this is an equivalence relation, that is it is reflexive, symmetric and transitive.

For R to be reflexive, (x, x) should be an element of R. That is,
x and x have the same number of pages.
Any book will have the same number of pages as itself, hence R is a reflexive relation.

For R to be symmetric, if (x, y) is an element of R then (y, x) should be an element of R. That is,
Let x and y have the same number of pages. Clearly, y and x will also have the same number of pages, hence R is a symmetric relation.

For R to be transitive, if (x, y) and (y, z) are elements of R then (x, z) should also be an element of R. That is,
Let x and y have the same number of pages and let y and z also have the same number of pages. It is obvious that x and z will also have the same number of pages. Hence, R is a transitive relation.
R is reflexive, symmetric and transitive, so it is an equivalence relation.

Note: It is important to check carefully for each condition. It is also recommended to check and verify each condition using a suitable example. Even if one case is false, the condition is not verified. Also if it is not possible to prove that relation is symmetric, reflexive or transitive, then use a suitable example to show that it is not.