Find the Domain and Range of the Relation on \(Z\)
Question
Let \(R\) be the relation on \(Z\) defined by
\[ R=\{(a,b):a,b\in Z,\ a-b \text{ is an integer}\} \]
Find the domain and range of \(R\).
Solution
Since \[ a,b\in Z, \] the difference \[ a-b \] is always an integer.
Hence every ordered pair of integers belongs to \(R\).
Therefore,
\[ R=Z\times Z \]
Domain = set of first elements
\[ \boxed{ Z } \]
Range = set of second elements
\[ \boxed{ Z } \]