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Show that each of the relation R in the set A = {x ∈ Z : 0 ≤ x ≤ 12}, given by R = {(a, b): |a – b| is a multiple of 4} is an equivalence relation. Find the set of all elements related to 1. |
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Answer» A = {x ∈ Z : 0 ≤ x ≤ 12} = {0,1, 2,3,4,5,6,7,8,9,10,11,12} and R = {(a, b): |a – b| is a multiple of 4} For any element a ∈ A, we have (a, a) ∈ R ⇒ |a – a| = 0 is a multiple of 4. ∴ R is reflexive. Now, let (a, b) ∈ R ⇒|a – b| is a multiple of 4. ⇒|–(a – b)| is a multiple of 4 ⇒|b – a| is a multiple of 4. ⇒ (b, a) ∈ R ∴ R is symmetric. Now, let (a, b), (b, c) ∈ R. ⇒|a – b| is a multiple of 4 and |b – c| is a multiple of 4. ⇒(a – b) is a multiple of 4 and (b – c) is a multiple of 4. ⇒(a – b + b – c) is a multiple of 4 ⇒(a – c) is a multiple of 4 ⇒|a – c| is a multiple of 4 ⇒ (a, c) ∈R ∴ R is transitive. Hence, R is an equivalence relation. The set of elements related to 1 is {1, 5, 9} since |1 – 1| = 0 is a multiple of 4 |5 – 1| = 4 is a multiple of 4 |9 – 1| = 8 is a multiple of 4 |
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