Question
If $$A = \left\{ {a,\,b,\,c,\,d} \right\},\,B = \left\{ {1,\,2,\,3} \right\},$$ which of the following sets of ordered pairs are not relations from $$A$$ to $$B\,?$$
A.
$$\left\{ {\left( {a,\,1} \right),\,\left( {a,\,3} \right)} \right\}$$
B.
$$\left\{ {\left( {b,\,1} \right),\,\left( {c,\,2} \right),\,\left( {d,\,1} \right)} \right\}$$
C.
$$\left\{ {\left( {a,\,2} \right),\,\left( {b,\,3} \right),\,\left( {3,\,b} \right)} \right\}$$
D.
$$\left\{ {\left( {a,\,1} \right),\,\left( {b,\,2} \right),\,\left( {c,\,3} \right)} \right\}$$
Answer :
$$\left\{ {\left( {a,\,2} \right),\,\left( {b,\,3} \right),\,\left( {3,\,b} \right)} \right\}$$
Solution :
$$\eqalign{
& \left\{ {\left( {a,\,1} \right),\,\left( {a,\,3} \right)} \right\} \subseteq A \times B\,\,\therefore \,{\text{This is a relation}} \cr
& \left\{ {\left( {b,\,1} \right),\,\left( {c,\,2} \right),\,\left( {d,\,1} \right)} \right\} \subseteq A \times B\,\,\therefore \,{\text{This is a relation}} \cr
& \left( {3,\,b} \right)\, \notin \,A \times B \cr
& \therefore \,\left\{ {\left( {a,\,2} \right),\,\left( {b,\,3} \right),\,\left( {3,\,b} \right)} \right\}{\text{ is not a relation from }}A{\text{ to }}B \cr
& \left\{ {\left( {a,\,1} \right),\,\left( {b,\,2} \right),\,\left( {c,\,3} \right)} \right\}\,\,{\text{is a relation from }}A{\text{ to }}B \cr} $$