Question
If $$A = \left\{ {1,\,2} \right\},\,B = \left\{ {1,\,3} \right\},$$ then $$\left( {A \times B} \right) \cup \left( {B \times A} \right)$$ is equal to :
A.
$$\left\{ {\left( {1,\,3} \right),\,\left( {2,\,3} \right),\,\left( {3,\,1} \right),\,\left( {3,\,2} \right),\,\left( {1,\,1} \right),\,\left( {2,\,1} \right),\,\left( {1,\,2} \right)} \right\}$$
B.
$$\left\{ {\left( {1,\,3} \right),\,\left( {3,\,1} \right),\,\left( {3,\,2} \right),\,\left( {2,\,3} \right)} \right\}$$
C.
$$\left\{ {\left( {1,\,3} \right),\,\left( {2,\,3} \right),\,\left( {3,\,1} \right),\,\left( {3,\,2} \right),\,\left( {1,\,1} \right)} \right\}$$
D.
None of these
Answer :
$$\left\{ {\left( {1,\,3} \right),\,\left( {2,\,3} \right),\,\left( {3,\,1} \right),\,\left( {3,\,2} \right),\,\left( {1,\,1} \right),\,\left( {2,\,1} \right),\,\left( {1,\,2} \right)} \right\}$$
Solution :
$$\eqalign{
& A \times B = \left\{ {1,\,2} \right\} \times \left\{ {1,\,3} \right\} = \left\{ {\left( {1,\,1} \right),\,\left( {1,\,3} \right),\,\left( {2,\,1} \right),\,\left( {2,\,3} \right)} \right\} \cr
& B \times A = \left\{ {1,\,3} \right\} \times \left\{ {1,\,2} \right\} = \left\{ {\left( {1,\,1} \right),\,\left( {1,\,2} \right),\,\left( {3,\,1} \right),\,\left( {3,\,2} \right)} \right\} \cr
& \therefore \,\,\left( {A \times B} \right) \cup \left( {B \times A} \right) \cr
& = \left\{ {\left( {1,\,1} \right),\,\left( {1,\,3} \right),\,\left( {2,\,1} \right),\,\left( {2,\,3} \right),\,\left( {1,\,2} \right),\,\left( {3,\,1} \right),\,\left( {3,\,2} \right)} \right\} \cr} $$