Prove that the relation R in the set A= \left \{ 1,2,3,4,5,6,7 \right \}  given by R= \left \{ \left ( a,b \right ):\left | a-b \right |is\; given \right \}  is an equivalence relation.

 

 

 

 
 
 
 
 

Answers (1)

Reflexive: \because \left | a-a \right |= 0\: which\; is \; even \; \therefore \left ( a,a \right )\epsilon R so ,R is reflexive.
Symmetric: Let \left ( a,b \right )\epsilon R  implies \left | a-b \right |  is even which means, \left | b-a \right |  is also even.That means, \left ( b,a \right )\epsilon R\cdot \; \therefore R\: is\: symmetric
Transitive:   Let \left ( a,b \right )\epsilon R     \left ( b,c \right )\epsilon R
It means \left | a-b \right |  and \left | b-c \right |   are even
That is , \left | a-b \right |= 2m\; \; \left | b-c \right |= 2n  where m,n\epsilon\: Z
\Rightarrow a-b= \pm 2m,\: \: b-c=\pm 2n
\Rightarrow a-b+b-c= a-c\pm 2\left ( m+n \right )
\Rightarrow \left | a-c \right |= 2\left ( m+n \right ), ie \left | a-c \right |is \: even\cdot
That is \left ( a,c \right )\epsilon R  so R is transitive
Hence R is equvivalence relation.

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