Question

# Let A={x ∈Z:0≤x≤12}, R={(a,b):a,b∈A,|a−b| is divisible by 4}, then which among the following options are correct

A
equivalence class of 1 is {1,5,9}
B
R is reflexive and symmetric but not transitive relation.
C
R is a Transitive relation
D
equivalence class of 1 is A

Solution

## The correct options are A equivalence class of 1 is {1,5,9} C R is a Transitive relationWe 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 a reflexive relation. Let (a,b)∈R ⇒|a−b|=4λ for some λ∈Z ⇒|b−a|=4λ1 for some λ1∈Z  [∵|a−b|=|b−a|] ⇒(b,a)∈R So, R is a symmetric relation. Let (a,b) and (b,c)∈R ⇒a−b=4α,b−c=4β,   α,β∈Z Now a−c=4(α+β) ⇒|a−c| is divisible by 4 ⇒(a,c)∈R  So, R is a transitive relation. Hence, R is an equivalence relation. Equivalance class of 1 is all possible values  of x such that (x,1)∈R,x∈A  ⇒|x−1| is divisible by 4 ⇒|x−1|=0,4,8 ⇒x−1=0,4,8 ⇒x=1,5,9 Thus, elements related to 1  are {1,5,9}.  Mathematics

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