Show That \(R\) is an Empty Relation from \(A\) into \(B\)
Question
Let \[ A=\{3,5\} \] and \[ B=\{7,11\}. \]
Let \[ R=\{(a,b):a\in A,\ b\in B,\ a-b \text{ is odd}\}. \]
Show that \(R\) is an empty relation from \(A\) into \(B\).
Solution
Elements of \(A\) and \(B\) are all odd numbers.
Difference of two odd numbers is always even.
\[ 3-7=-4,\quad 3-11=-8 \]
\[ 5-7=-2,\quad 5-11=-6 \]
All differences are even, not odd.
Therefore, there is no ordered pair satisfying the condition.
Hence,
\[ \boxed{ R=\phi } \]
Thus, \(R\) is an empty relation from \(A\) into \(B\).