Question
$$\eqalign{
& {\text{Let }}R = \left\{ {x|x\, \in \,N,\,x{\text{ is a multiple of 3 and }}x \leqslant 100} \right\} \cr
& \,\,\,\,\,\,\,\,\,\,S = \left\{ {x|x\, \in \,N,\,x{\text{ is a multiple of 5 and }}x \leqslant 100} \right\} \cr} $$
What is the number of elements in $$\left( {R \times S} \right) \cap \left( {S \times R} \right)$$
A.
36
B.
33
C.
20
D.
6
Answer :
36
Solution :
$$\eqalign{
& {\text{Let }}R = \left\{ {x:x\, \in \,N,\,x{\text{ is a multiple of 3 and }}x \leqslant 100} \right\} \cr
& {\text{and}}\,S = \left\{ {x:x\, \in \,N,\,x{\text{ is a multiple of 5 and }}x \leqslant 100} \right\} \cr
& \therefore \,R = \left\{ {3,\,6,\,9,\,12,\,15,\,.....,\,99} \right\} \cr
& {\text{and}}\,S = \left\{ {5,\,10,\,15,.....,95,\,100} \right\} \cr
& {\text{Now, }}\left( {R \times S} \right) \cap \left( {S \times R} \right) \cr
& = \left( {R \cap S} \right) \times \left( {S \cap R} \right) \cr
& = \left( {15,\,30,\,45,\,60,\,75,\,90} \right) \times \left( {15,\,30,\,45,\,60,\,75,\,90} \right) \cr
& \therefore {\text{ Number of elements in }}\left( {R \times S} \right) \cap \left( {S \times R} \right) = 6 \times 6 = 36 \cr} $$