Ex 1.1, 10 - Given an example of a relation. Which is - Chapter 1 - To prove relation reflexive/trasitive/symmetric/equivalent

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  1. Chapter 1 Class 12 Relation and Functions
  2. Serial order wise
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Ex 1.1, 10 Given an example of a relation. Which is (i) Symmetric but neither reflexive nor transitive. Let A = {1, 2, 3}. Let relation R on set A be Let R = {(1, 2), (2, 1)} Check Reflexive If the relation is reflexive, then (a, a) ∈ R for every a ∈ {1,2,3} Since (1, 1), (2, 2), (3, 3) ∉ R ∴ R is not reflexive Check Symmetric Since (1, 2) ∈ R , (2, 1) ∈ R So, If (a, b) ∈ R, then (b, a) ∈ R ∴ R is symmetric. Check transitive To check whether transitive or not, If (a, b) ∈ R & (b, c) ∈ R , then (a, c) ∈ R If a = 1, b = 2, but there is no c (no third element) Similarly, if a = 2, b = 1, but there is no c (no third element) Hence ,R is not transitive Hence, relation R is symmetric but not reflexive and transitive Ex 1.1,10 Given an example of a relation. Which is (ii) Transitive but neither reflexive nor symmetric. Let R = {(a, b): a < b} Check reflexive Since a cannot be less than a a ≮ a So, (a, a) ∉ R ∴ R is not reflexive. Check symmetric If a < b , then b cannot be less than a i.e. b ≮ a So, if (a, b) ∈ R , (b, a) ∉ R ∴ R is not symmetric Check transitive If a < b & b < c, then a < c So, if (a, b) ∈ R, (b, c) ∈ R, then (a, c) ∈ R ∴ R is transitive. Hence, relation R is transitive but not reflexive and symmetric. Ex 1.1,10 Given an example of a relation. Which is (iii) Reflexive and symmetric but not transitive. Let A = {1, 2, 3}. Let relation R on set A be Let R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 1), (1, 3), (3, 1)} Check Reflexive If the relation is reflexive, then (a, a) ∈ R for every a ∈ {1,2,3} Since (1, 1), (2, 2), (3, 3) ∈ R ∴ R is reflexive Check Symmetric To check whether symmetric or not, If (a, b) ∈ R, then (b, a) ∈ R Since, If (1, 2) ∈ R , then (2, 1) ∈ R & if (1, 3) ∈ R , then (3, 1) ∈ R ∴ R is symmetric. Check transitive To check whether transitive or not, If (a,b) ∈ R & (b,c) ∈ R , then (a,c) ∈ R Here, a = 1, b = 2 or 3, but there is no c (no third element) Hence ,R is not transitive Hence, relation R is reflexive and symmetric but not transitive. Ex 1.1, 10 Given an example of a relation. Which is (iv) Reflexive and transitive but not symmetric. Let A = {1, 2, 3}. Let relation R on set A be Let R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 3), (1, 3)} Check Reflexive If the relation is reflexive, then (a, a) ∈ R for every a ∈ {1,2,3} Since (1, 1), (2, 2), (3, 3) ∈ R ∴ R is reflexive Check Symmetric Since (1, 2) ∈ R , but (2, 1) ∉ R If (a, b) ∈ R, then (b, a) ∉ R ∴ R is not symmetric. Check transitive Since (1, 2) ∈ R , (2, 3) ∈ R & (1, 3) ∈ R So, If (a, b) ∈ R , (b, c) ∈ R , then (a, c) ∈ R ∴ R is transitive. Hence, relation R is reflexive and transitive but not symmetric. Ex 1.1, 10 Given an example of a relation. Which is (v) Symmetric and transitive but not reflexive. Let A = {1, 2, 3}. Let relation R on set A be Let R = {(1, 2), (2, 1), (1, 3), (3, 1), (2, 3), (3, 2)} Check Reflexive If the relation is reflexive, then (a, a) ∈ R for every a ∈ {1,2,3} Since (1, 1), (2, 2), (3, 3) ∉ R ∴ R is not reflexive Check Symmetric Since (1, 2) ∈ R , (2, 1) ∈ R & (1, 3) ∈ R , (3, 1) ∈ R & (2, 3) ∈ R , (3, 2) ∈ R So, If (a, b) ∈ R, then (b, a) ∈ R ∴ R is symmetric. Check transitive Since (1, 2) ∈ R , (2, 3) ∈ R & (1, 3) ∈ R & (2, 1) ∈ R , (1, 3) ∈ R & (2, 3) ∈ R & (3, 1) ∈ R , (1, 2) ∈ R & (3, 2) ∈ R So, If (a, b) ∈ R , (b, c) ∈ R , then (a, c) ∈ R ∴ R is transitive. Hence, relation R is symmetric and transitive but not reflexive

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