Problem 4 Solution

This page contains the NCERT mathematics class 12 chapter Relations and Functions Exercise 1.1 Problem 4 Solution. Solutions for other problems are available at Exercise 1.1 Solutions
Exercise 1.1 Problem 4 Solution
4. Show that the relation R in R defined as R = (a, b) : a \le b, is reflexive and transitive but not symmetric.
To Check whether R is Reflexive: The relation R in the set A is reflexive if (a, a)R, for every aA.
As every element a is equal to itself, it satisfies the condition {a \le a}.
(a, a)R
R is Reflexive
To Check whether R is Symmetric: The relation R in the set A is symmetric if (a_1, a_2)R implies that (a_2, a_1)R, for all a_1, a_2A
We have 2 ≤ 4. But 4 \nleq 2 (4 > 2). Thus for many elements when (a, b)R, we have (b, a)R.
R is not symmetric.
To Check whether R is Transitive: The relation R in the set A is transitive if (a_1, a_2)R and (a_2, a_3)R implies that (a_1, a_3)R, for all a_1, a_2, a_3A
When {a \le b} and {b \le c} then we always have {a \le c}
R is Transitive
R is both reflexive and transitive but not symmetric