Question

Show that the relation R in the set A of all books in a library of a school given given by R = {(x,y) : x and y have the sender pages } is an equivalence relation.​

Answer 1

Given ,

A = { All the books in a library of a school}

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

1) Reflexivity : Any relation which exhibit the reflexive property, is said to be reflexive. In this relation,  every element involved is in relation with itself.

Here,

(x,x) ∈ R ⇒ R is reflexive on A.

Symmetric : In a symmetric relation , x and y can be interchanged i.e,, if ordered pair (x,y) is an element of the relation, the ordered pair (y,x)  is an also the element of the relation.

Here,

Since books x and y have the same pages , so (x,y) ∈ R.

Since books y and x have the same pages means (y,x) ∈ R.

⇒ R is symmetric on A.

Transitive :  

Books x,y and z have the same number of pages .

(x,y)∈ R and (y,z)∈ R .

⇒ (x,z) ∈ R .

⇒ R is transitive on A

Hence ,R is an equivalence relation.