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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.

Class 12, NCERT Chapter 1,  Exercise 1.1, Q7 Set  A  is the set of all books in the library of a college. Given, R = { x ,  y ):  x  and  y  have the same number of pages} Now, R is reflexive since ( x ,  x ) ∈ R as  x  and  x  has the same number of pages. Let ( x ,  y ) ∈ R  ⇒  x  and  y  have the same number of pages. ⇒  y  and  x  have the same number of pages. ⇒ ( y ,  x ) ∈ R ∴R is symmetric. Now, let ( x ,  y ) ∈R and ( y ,  z ) ∈ R. ⇒  x  and  y  and 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 ∴R is transitive. Hence, R is an equivalence relation.

Let A be the set of all 50 students of Class X in a school. Let f : A → N be function defined by f(x) = roll number of the student x. Show that f is one-one but not onto.

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