Example 23 - Chapter 1 Class 12 Relation and Functions
Last updated at April 16, 2024 by Teachoo
Finding number of relations
Finding number of relations
Last updated at April 16, 2024 by Teachoo
Example 23 Let A = {1, 2, 3}. Then show that the number of relations containing (1, 2) and (2, 3) which are reflexive and transitive but not symmetric is three. Total possible pairs = {(1, 1) , (1, 2), (1, 3), (2, 1) , (2, 2), (2, 3), (3, 1) , (3, 2), (3, 3) } Each relation should have (1, 2) and (2, 3) in it For other pairs, Let’s check which pairs will be in relation, and which won’t be Total possible pairs = { (1, 1) , (1, 2), (1, 3), (2, 1) , (2, 2), (2, 3), (3, 1) , (3, 2), (3, 3) } Reflexive means (a, a) should be in relation . So, (1, 1) , (2, 2) , (3, 3) should be in a relation Symmetric means if (a, b) is in relation, then (b, a) should be in relation . We need relation which is not symmetric. So, since (1, 2) is in relation, (2, 1) should not be in relation & since (2, 3) is in relation, (3, 2) should not be in relation Transitive means if (a, b) is in relation, & (b, c) is in relation, then (a, c) is in relation So, if (1, 2) is in relation, & (2, 3) is in relation, then (1, 3) should be in relation Relation R1 = { Total possible pairs = { (1, 1) , (1, 2), (1, 3), (2, 1) , (2, 2), (2, 3), (3, 1) , (3, 2), (3, 3) } So, smallest relation is R1 = { (1, 2), (2, 3), (1, 1), (2, 2), (3, 3), (1, 3) } Checking more relations We cannot add both (2, 1) & (3, 2) together as it is not symmetric R = { (1, 2), (2, 3), (1, 1), (2, 2), (3, 3), (1, 3) , (2, 1), (3, 2)} If we add only (3, 1) to R1 R = { (1, 2), (2, 3), (1, 1), (2, 2), (3, 3), (1, 3), (3, 1) } R is reflexive but not symmetric & transitive. So, not possible If we add only (2, 1) to R1 R2 = { (1, 2), (2, 3), (1, 1), (2, 2), (3, 3), (1, 3), (2, 1) } R2 is reflexive, transitive but not symmetric If we add only (3, 2) to R1 R3 = { (1, 2), (2, 3), (1, 1), (2, 2), (3, 3), (1, 3), (3, 2) } R3 is reflexive, transitive but not symmetric Hence, there are only three possible relations R1 = { (1, 2), (2, 3), (1, 1), (2, 2), (3, 3), (1, 3) } R2 = { (1, 2), (2, 3), (1, 1), (2, 2), (3, 3), (1, 3), (2, 1)} R3 = { (1, 2), (2, 3), (1, 1), (2, 2), (3, 3), (1, 3), (3, 2)}