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Let $R=\{x \mid x \in N, x$ is a multiple of 3 and $x \leq 100\}$ $S=\{x \mid x \in N, x$ is a multiple of 5 and $x \leq 100\}$
What is the number of elements in $(R \times S) \cap(S \times R)^{2}$
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What is the number of elements in $(R \times S) \cap(S \times R)^{2}$
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The correct answer is:
36
Let $R=\{x: x \in N, x$ is a multiple of 3 and $x \leq 100\}$
and $S=\{x: x \in N, x$ is a multiple of 5 and $x \leq 100\}$ $R=\{3,6,9,12,15, \ldots, 99\}$
and $S=\{5,10,15, \ldots, 95,100\}$
Now, $(R \times S) \cap(S \times R)=(R \cap S) \times(S \cap R)$
$=(15,30,45,60,75,90) \times(15,30,45,60,75,90)$
Number of elements in $(R \times S) \cap(S \times R)$
$=6 \times 6=36$
and $S=\{x: x \in N, x$ is a multiple of 5 and $x \leq 100\}$ $R=\{3,6,9,12,15, \ldots, 99\}$
and $S=\{5,10,15, \ldots, 95,100\}$
Now, $(R \times S) \cap(S \times R)=(R \cap S) \times(S \cap R)$
$=(15,30,45,60,75,90) \times(15,30,45,60,75,90)$
Number of elements in $(R \times S) \cap(S \times R)$
$=6 \times 6=36$
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