Class JEE Mathematics Sets, Relations, and Functions Q #1070
KNOWLEDGE BASED
APPLY
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) $5^2$
Prev Next
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 $5$.

More from this Chapter

MCQ_SINGLE
Let $A = {x \in (0, \pi) - {\frac{\pi}{2}} : \log_{(2/\pi)} |\sin x| + \log_{(2/\pi)} |\cos x| = 2}$ and $B = {x \ge 0 : x(x-4) - 3|x-2| + 6 = 0}$. Then $n(A \cup B)$ is equal to :
MCQ_SINGLE
Let $R = \{(3, 3), (6, 6), (9, 9), (12, 12), (6, 12), (3, 9), (3, 12), (3, 6)\}$ be a relation on the set $A = \{3, 6, 9, 12\}$. The relation is :
MCQ_SINGLE
Let a relation $R$ on $N \times N$ be defined as: $(x_1, y_1) R (x_2, y_2)$ if and only if $x_1 \le x_2$ or $y_1 \le y_2$. Consider the two statements: (I) $R$ is reflexive but not symmetric. (II) $R$ is transitive Then which one of the following is true?
MCQ_SINGLE
Let R be a relation from the set ${1, 2, 3, …, 60}$ to itself such that $R = {(a, b) : b = pq}$, where $p, q \geqslant 3$ are prime numbers}. Then, the number of elements in R is :
MCQ_SINGLE
Let $A = \{ (\alpha, \beta) \in R \times R : |\alpha - 1| \leq 4 \text{ and } |\beta - 5| \leq 6 \}$ and $B = \{ (\alpha, \beta) \in R \times R : 16(\alpha - 2)^2 + 9(\beta - 6)^2 \leq 144 \}$. Then
View All Questions