Ex 1.1, 8 - In set A = {1, 2, 3, 4, 5}, R = {(a, b): |a-b| - Ex 1.1

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  1. Chapter 1 Class 12 Relation and Functions
  2. Serial order wise

Transcript

Ex 1.1, 8 (Introduction) Show that the relation R in the set A = {1, 2, 3, 4, 5} given by R = {(a,b):|a b| is even} , is an equivalence relation. Show that all the elements of {1, 3, 5} are related to each other and all the elements of {2, 4} are related to each other. But no element of {1, 3, 5} is related to any element of {2, 4}. Modulus function 1 = 1 2 = 2 0 = 0 1 = 1 3 = 3 Ex 1.1, 8 Show that the relation R in the set A = {1, 2, 3, 4, 5} given by R = {(a,b):|a b| is even} , is an equivalence relation. Show that all the elements of {1, 3, 5} are related to each other and all the elements of {2, 4} are related to each other. But no element of {1, 3, 5} is related to any element of {2, 4}. R = {(a, b):|a b| is even} where a, b A Check Reflexive Since |a a| = |0| = 0 & 0 is always even |a a| is even (a, a) R, R is reflexive. Check symmetric We know that |a b| = |b a| Hence, if |a b| is even, then |b a| is also even Hence, If (a, b) R, then (b, a) R R is symmetric Check transitive If |a b| is even , then (a b) is even Similarly, if |b c| is even , then (b c) is even Now, Sum of even numbers is also even a b + b c is even a c is even Hence, |a c| is even So, If |a b| &|b c| is even , then |a c| is even i.e. If (a, b) R & (b, c) R , then (a, c) R R is transitive Since R is reflexive, symmetric and transitive, it is equivalence relation R = {(a,b):|a b| is even} Show that all the elements of {1, 3, 5} are related to each other and all the elements of {2, 4} are related to each other. But no element of {1, 3, 5} is related to any element of {2, 4}. In {1, 3, 5}, All elements are odd, So, difference between any 2 numbers is always even Hence, Modulus of difference between any 2 numbers is always even Hence, element of {1, 3, 5} are related to each other In {2, 4}, All elements are even, So, difference between any 2 numbers is always even Hence, Modulus of difference between any 2 numbers is always even Hence, element of {2, 4} are related to each other In {1, 3, 5} & {2, 4}, Elements of {1, 3, 5} are odd Elements of {2, 4} are even Difference of one element from {1, 3, 5} and one element from {2, 4} is odd As Difference of even and odd number is always odd Difference is not even If difference not even, Modulus of difference also not even Hence, element of {1, 3, 5} & {2, 4} are not related to each other

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