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Question

# Let A = {x ∈ Z:0≤x≤12}. Show that R={(a,b):a,b∈A,|a−b| is divisible by 4} is an equivalence relation. Find the set of all elements related to 1. Also write the equivalence class [2].

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Solution

## We have R={(a,b):a,b∈A,|a−b| is divisible by 4}, where a,b∈{0,1,2,3,....,12}. For any a∈ A, we have |a−a|=0, which is divisible by 4 ⇒(a,a)∈ R. So, R is reflexive. For any (a,b)∈R |a−b| is divisible by 4 ⇒|a−b|=4λ for some λ ∈ N ⇒|b−a|=4λ for some λ ∈ N [∵|a−b|=|b−a|] ⇒(b,a)∈ R So, R is symmetric. Let (a,b)∈R and (b,c)∈R, then |a−b| is divisible by 4 and |b−c| is divisible by 4 ⇒|a−b|=4λ and |b−c|=4μ ∵a−b and b−c are both multiples of 4 ∴a−b+b−c=a−c is a multiple of 4 ⇒|a−c| is divisible by 4 ⇒(a,c)∈ R So, R is transitive. Hence, R is an equivalence relation. Let x be an element of A such that (x,1)∈R, then |x−1| is divisible by 4 ⇒|x−1|=0,4,8,12 ⇒x−1=0,4,8,12 ⇒x=1,5,9 Thus, elements related to 1 are {1,5,9}. Also, |x−2| is divisible by 4 ⇒|x−2|=0,4,8,12 ⇒x−2=0,4,8,12 ⇒x=2,6,10 Hence, the equivalence class [2]={2,6,10}

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