Example1 Let A be the set of all students of a boys school. Show that the relation R in A given by R = {(a, b): a is the sister of b} is the empty relation and R′ = {(a, b): the difference between heights of a and b is less than 3 meters} is the universal relation.

RELATIONS AND FUNCTIONS

Class 12, NCERT Chapter1, Example1

Solution

Since the school is a boy's school, no student at the school can be the sister of any student in the school. Hence, R = φ, showing that R is the empty relation. It is also obvious that the difference between the heights of any two students of the school has to be less than 3 meters. This shows that R′ = A × A is a universal relation.

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Example2
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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.

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