Which of the Following are Relations from \(A\) to \(B\)?
Question
If \[ A=\{1,2,3\},\quad B=\{4,5,6\}, \] which of the following are relations from \(A\) to \(B\)? Give reasons.
(i) \[ \{(1,6),(3,4),(5,2)\} \]
(ii) \[ \{(1,5),(2,6),(3,4),(3,6)\} \]
(iii) \[ \{(4,2),(4,3),(5,1)\} \]
(iv) \[ A\times B \]
Solution
A relation from \(A\) to \(B\) is a subset of \[ A\times B. \]
(i)
\[ (5,2) \] is not in \[ A\times B \] because \[ 5\notin A \] and \[ 2\notin B. \]
Hence, it is not a relation.
(ii)
All first elements belong to \(A\) and all second elements belong to \(B\).
Hence, it is a relation from \(A\) to \(B\).
(iii)
\[ 4,5\notin A \] and \[ 2,3,1\notin B. \]
Hence, it is not a relation.
(iv)
\[ A\times B \] is always a relation from \(A\) to \(B\).
Hence, it is a relation from \(A\) to \(B\).
Therefore,
\[ \boxed{ \text{(ii) and (iv) are relations from } A \text{ to } B. } \]