Write R in Roster Form if x − y is Odd
Question
Let
\[ A=\{1,2,3,5\}, \quad B=\{4,6,9\} \]
and \( R \) be a relation from \( A \) to \( B \) defined by
\[ R=\{(x,y): x-y \text{ is odd}\} \]
Write \( R \) in roster form.
Solution
\[ 1-4=-3 \text{ (odd)},\quad 1-6=-5 \text{ (odd)} \]
\[ 1-9=-8 \text{ (even)} \]
\[ 2-4=-2 \text{ (even)},\quad 2-6=-4 \text{ (even)} \]
\[ 2-9=-7 \text{ (odd)} \]
\[ 3-4=-1 \text{ (odd)},\quad 3-6=-3 \text{ (odd)} \]
\[ 3-9=-6 \text{ (even)} \]
\[ 5-4=1 \text{ (odd)},\quad 5-6=-1 \text{ (odd)} \]
\[ 5-9=-4 \text{ (even)} \]
Therefore,
\[ R=\{(1,4),(1,6),(2,9),(3,4),(3,6),(5,4),(5,6)\} \]
Hence,
\[ \boxed{\{(1,4),(1,6),(2,9),(3,4),(3,6),(5,4),(5,6)\}} \]