Check sibling questions

Example 47 - Let A = {1, 2, 3}. Show that number of relations

Example 47 - Chapter 1 Class 12 Relation and Functions - Part 2
Example 47 - Chapter 1 Class 12 Relation and Functions - Part 3 Example 47 - Chapter 1 Class 12 Relation and Functions - Part 4 Example 47 - Chapter 1 Class 12 Relation and Functions - Part 5


Transcript

Example 47 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)}

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Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.