Q.1Determine whether each of the following relations are reflexive, symmetric and
transitive:

(iv). Relation R in the set Z of all integers defined as R = \{(x,y): x - y \;is\;an\;integer\}

Answers (1)

R = \{(x,y): x - y \;is\;an\;integer\} 

For x \in Z\left ( x,x \right ) \in R as x-x = 0 which is an integer.

So,it is reflxive.

 For x,y \in Z ,\left ( x,y \right ) \in R and \left ( y,x \right ) \in R because x-y \, \, and \, \, y-x  are both integers.

So, it is symmetric.

 For x,y,z \in Z ,\left ( x,y \right ),\left ( y,z \right ) \in R as x-y \, \, and \, \, y-z are both integers.

Now, x-z = \left ( x-y \right )+\left ( y-z \right ) is also an integer.

So,\left ( x,z \right ) \in R and hence it is transitive.

Hence, it is  reflexive, symmetric and
transitive.

 

 

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