Relation \(R\) Defined by \(y=x+5\)
Question
Define a relation \(R\) on the set \(N\) of natural numbers by
\[ R=\{(x,y):y=x+5,\ x \text{ is a natural number less than }4,\ x,y\in N\} \]
Depict this relation using:
(i) roster form
(ii) an arrow diagram
Write down the domain and range of \(R\).
Solution
Since \(x\) is a natural number less than \(4\),
\[ x=1,2,3 \]
Using \[ y=x+5 \]
\[ x=1 \Rightarrow y=6 \]
\[ x=2 \Rightarrow y=7 \]
\[ x=3 \Rightarrow y=8 \]
(i) Roster Form
\[ \boxed{ R=\{(1,6),(2,7),(3,8)\} } \]
(ii) Arrow Diagram
1 → 6
2 → 7
3 → 8
2 → 7
3 → 8
Domain:
\[ \boxed{ \{1,2,3\} } \]
Range:
\[ \boxed{ \{6,7,8\} } \]