Q193 / 270 · NDA 2025 (Phase 2) · Mathematics
Let $A = \{-3, -2, -1, 0, 1, 2, 3\}$ and $B = \{0, 1, 4, 9\}$. How many elements does the subset of $A \times B$ corresponding to the relation $R = \{(x, y) : |x| < y\}$ have, where $x \in A$ and $y \in B$?
- A$9$
- B$12$
- Correct answerC$15$
- D$16$
Solution
Given $A = \{-3, -2, -1, 0, 1, 2, 3\}$ and $B = \{0, 1, 4, 9\}$.
The relation is $R = \{(x, y) : |x| < y, x \in A, y \in B\}$.
We evaluate the number of valid $x \in A$ for each $y \in B$:
For $y = 0$, $|x| < 0 \Rightarrow$ no such $x \in A$. Number of pairs is $0$.
For $y = 1$, $|x| < 1 \Rightarrow x = 0$. Number of pairs is $1$.
For $y = 4$, $|x| < 4 \Rightarrow x \in \{-3, -2, -1, 0, 1, 2, 3\}$. Number of pairs is $7$.
For $y = 9$, $|x| < 9 \Rightarrow x \in \{-3, -2, -1, 0, 1, 2, 3\}$. Number of pairs is $7$.
Total number of elements in $R = 0 + 1 + 7 + 7 = 15$.
Answer: $15$
The relation is $R = \{(x, y) : |x| < y, x \in A, y \in B\}$.
We evaluate the number of valid $x \in A$ for each $y \in B$:
For $y = 0$, $|x| < 0 \Rightarrow$ no such $x \in A$. Number of pairs is $0$.
For $y = 1$, $|x| < 1 \Rightarrow x = 0$. Number of pairs is $1$.
For $y = 4$, $|x| < 4 \Rightarrow x \in \{-3, -2, -1, 0, 1, 2, 3\}$. Number of pairs is $7$.
For $y = 9$, $|x| < 9 \Rightarrow x \in \{-3, -2, -1, 0, 1, 2, 3\}$. Number of pairs is $7$.
Total number of elements in $R = 0 + 1 + 7 + 7 = 15$.
Answer: $15$
Hence, the correct answer is option: 'C'.