Question

R be a relation ob Z defined by R={(a,b):a-b is an integer show that R is an equivalence relation​

Answer 1

$\textbf{Given:}$

R defined on Z by

$aRb\;{\iff}\;\text{a-b is an integer}$

$\textbf{To prove:}$

$\text{R is an equivalence relation}$

$\textbf{Solution:}$

$\text{Let a{\in}Z }$

$\implies\,a-a=0\;\text{which is an integer}$

$\implies\,aRa$

$\therefore\,\textbf{R is reflexive}$

$\text{Let aRb }$

$\implies\,a-b\;\text{is an integer}$

$\implies\,b-a\;\text{is also an integer}$

$\implies\,bRa$

$\therefore\,\textbf{R is symmetric}$

$\text{Let aRb and bRc }$

$\text{Then,}\;\;a-b=k\;\text{and}\;b-c=l\;\;\text{where k,l are integers}$

$\text{Adding we get,}$

$(a-b)+(b-c)=k+l$

$\implies\,a-c=k+l\;\text{which is an integer}$

$\implies\,aRc$

$\therefore\textbf{R is transitive}$

$\textbf{Hence R is an equivalence relation}$

Find more:

show that the relation R defined on the set A={1, 2} as R={(1, 1), (2, 2), (1, 2) } is not symmetric

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