Class JEE Mathematics Sets, Relations, and Functions Q #1070
KNOWLEDGE BASED
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4 Marks 2012 AIEEE 2012 MCQ SINGLE
Let $X = {1, 2, 3, 4, 5}$. The number of different ordered pairs $(Y, Z)$ that can be formed such that $Y \subseteq X$, $Z \subseteq X$ and $Y \cap Z$ is empty, is:
(A) $3^5$
(B) $2^5$
(C) $5^3$
(D) $5^2$
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Correct Answer: A
Explanation
For any element $x_i$ present in $X$, 4 cases arise while making subsets $Y$ and $Z$.

Case 1: $x_i \in Y, x_i \in Z \implies Y \cap Z \neq \emptyset$

Case 2: $x_i \in Y, x_i \notin Z \implies Y \cap Z = \emptyset$

Case 3: $x_i \notin Y, x_i \in Z \implies Y \cap Z = \emptyset$

Case 4: $x_i \notin Y, x_i \notin Z \implies Y \cap Z = \emptyset$

Therefore, for every element, the number of ways is $3$ for which $Y \cap Z = \emptyset$.

Thus, the total number of ways is $3 \times 3 \times 3 \times 3 \times 3 = 3^5$ since the number of elements in set $X$ is $5$.

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