Show that each of the relation R in the set, given by (i) (ii) is an equivalence relation. Find the set of all elements related to 1 in each case.

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(i) For any element a ∈A, we have (a, a) ∈ R as is a multiple of 4. ∴R is reflexive. Now, let (a, b) ∈ R ⇒ is a multiple of 4. ⇒ (b, a) ∈ R ∴R is symmetric. Now, let (a, b), (b, c) ∈ R. ⇒ (a, c) ∈R ∴ R is transitive. Hence, R is an...
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(i) For any elementa∈A, we have (a,a) ∈ R asis a multiple of 4. ∴R is reflexive. Now, let (a,b) ∈ R ⇒is a multiple of 4. ⇒ (b,a) ∈ R ∴R is symmetric. Now, let (a,b), (b,c) ∈ R. ⇒ (a,c) ∈R ∴ R is transitive. Hence, R is an equivalence relation. The set of elements related to 1 is {1, 5, 9} since (ii) R = {(a,b):a=b} For any elementa∈A, we have (a,a) ∈ R, sincea=a. ∴R is reflexive. Now, let (a,b) ∈ R. ⇒a=b ⇒b=a ⇒ (b,a) ∈ R ∴R is symmetric. Now, let (a,b) ∈ R and (b,c) ∈ R. ⇒a=bandb=c ⇒a=c ⇒ (a,c) ∈ R ∴ R is transitive. Hence, R is an equivalence relation. The elements in R that are related to 1 will be those elements from set A which are equal to 1. Hence, the set of elements related to 1 is {1}. read less
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