If A = {3, 5, 6, 9} and R = {(x, y) : x + y < 18}, Find R in Roster Form
Question
If
\[ A=\{3,5,6,9\} \]
and \( R \) is a relation in \( A \) defined as
\[ R=\{(x,y): x+y<18\} \]
then \( R \) in roster form is ……………………….
Solution
We check all ordered pairs \( (x,y) \) where:
\[ x,y\in A \]
and
\[ x+y<18 \]
Given:
\[ A=\{3,5,6,9\} \]
Now test each pair:
\[ (3,3): 3+3=6<18 \]
\[ (3,5): 3+5=8<18 \]
\[ (3,6): 3+6=9<18 \]
\[ (3,9): 3+9=12<18 \]
\[ (5,3): 5+3=8<18 \]
\[ (5,5): 5+5=10<18 \]
\[ (5,6): 5+6=11<18 \]
\[ (5,9): 5+9=14<18 \]
\[ (6,3): 6+3=9<18 \]
\[ (6,5): 6+5=11<18 \]
\[ (6,6): 6+6=12<18 \]
\[ (6,9): 6+9=15<18 \]
\[ (9,3): 9+3=12<18 \]
\[ (9,5): 9+5=14<18 \]
\[ (9,6): 9+6=15<18 \]
\[ (9,9): 9+9=18 \not<18 \]
So all pairs except \( (9,9) \) belong to the relation.
Therefore,
\[ R= \{ (3,3),(3,5),(3,6),(3,9), \]
\[ (5,3),(5,5),(5,6),(5,9), \]
\[ (6,3),(6,5),(6,6),(6,9), \]
\[ (9,3),(9,5),(9,6) \} \]
Hence,
\[ \boxed{ R= \{ (3,3),(3,5),(3,6),(3,9), (5,3),(5,5),(5,6),(5,9), (6,3),(6,5),(6,6),(6,9), (9,3),(9,5),(9,6) \} } \]