Let A = {-3, -2, -1, 0, 1, 2, 3}. Let R be a relation on A defined by xRy if and only if $0\le x^2+2y\le 4$. Let $l$ be the number of elements in R and $m$ be the minimum number of elements required to be added in R to make it a reflexive relation. Then $l + m$ is equal to
18
Calculating Relation R Elements and Reflexivity ($l+m$)
Given set A = {-3, -2, -1, 0, 1, 2, 3} and relation R defined by $xRy \iff 0 \le x^2 + 2y \le 4$. We need to find $l$, the number of elements in R, and $m$, the minimum elements to add for R to be reflexive, then calculate $l + m$.
Finding Relation R Elements ($l$)
We check the condition $0 \le x^2 + 2y \le 4$ for all pairs $(x, y)$ where $x, y \in A$. This involves finding the range for $y$ for each $x$ in A that satisfies the inequalities.
| $x$ | $x^2$ | Condition $0 \le x^2+2y \le 4$ | Range for $y$ | Valid $y$ in A | Count | Pairs in R |
|---|---|---|---|---|---|---|
| -3 | 9 | $0 \le 9+2y \le 4$ | $y \le -2.5$ | {-3} | 1 | (-3, -3) |
| -2 | 4 | $0 \le 4+2y \le 4$ | $-2 \le y \le 0$ | {-2, -1, 0} | 3 | (-2, -2), (-2, -1), (-2, 0) |
| -1 | 1 | $0 \le 1+2y \le 4$ | $-0.5 \le y \le 1.5$ | {0, 1} | 2 | (-1, 0), (-1, 1) |
| 0 | 0 | $0 \le 0+2y \le 4$ | $0 \le y \le 2$ | {0, 1, 2} | 3 | (0, 0), (0, 1), (0, 2) |
| 1 | 1 | $0 \le 1+2y \le 4$ | $-0.5 \le y \le 1.5$ | {0, 1} | 2 | (1, 0), (1, 1) |
| 2 | 4 | $0 \le 4+2y \le 4$ | $-2 \le y \le 0$ | {-2, -1, 0} | 3 | (2, -2), (2, -1), (2, 0) |
| 3 | 9 | $0 \le 9+2y \le 4$ | $y \le -2.5$ | {-3} | 1 | (3, -3) |
Summing the counts for each $x$: $l = 1 + 3 + 2 + 3 + 2 + 3 + 1 = 15$.
Determining Minimum Elements for Reflexivity ($m$)
A relation R is reflexive if $(x, x) \in R$ for every $x \in A$. We test the condition $0 \le x^2 + 2x \le 4$ for each $x \in A$ to see if the diagonal pairs are present.
- For $x = -3$: $x^2+2x = 3$. $0 \le 3 \le 4$ is true. $(-3, -3) \in R$.
- For $x = -2$: $x^2+2x = 0$. $0 \le 0 \le 4$ is true. $(-2, -2) \in R$.
- For $x = -1$: $x^2+2x = -1$. $0 \le -1 \le 4$ is false. $(-1, -1) \notin R$.
- For $x = 0$: $x^2+2x = 0$. $0 \le 0 \le 4$ is true. $(0, 0) \in R$.
- For $x = 1$: $x^2+2x = 3$. $0 \le 3 \le 4$ is true. $(1, 1) \in R$.
- For $x = 2$: $x^2+2x = 8$. $0 \le 8 \le 4$ is false. $(2, 2) \notin R$.
- For $x = 3$: $x^2+2x = 15$. $0 \le 15 \le 4$ is false. $(3, 3) \notin R$.
The pairs needed for reflexivity that are missing from R are $(-1, -1)$, $(2, 2)$, and $(3, 3)$.
Therefore, the minimum number of elements to add to make R reflexive is $m = 3$.
Final Calculation ($l + m$)
We have calculated $l = 15$ (the number of elements in R) and $m = 3$ (the minimum elements needed for reflexivity).
The sum $l + m$ is $15 + 3 = 18$.
Similar Questions
Let S be the set of the first 11 natural numbers. Then the number of elements in $A = \{B \subseteq S : n(B) \geq 2$ and the product of all elements of B is even is ________.
- The number of $3 \times 2$ matrices A, which can be formed using the elements of the set $\{-2, -1, 0, 1, 2\}$ such that the sum of all the diagonal elements of $A^T A$ is 5, is ______
- The number of numbers greater than 5000, less than 9000 and divisible by 3, that can be formed using the digits 0, 1, 2, 5, 9, if the repetition of the digits is allowed, is ______
- Let $z = (1 + i)(1 + 2i)(1 + 3i) \dots (1 + ni)$, where $i = \sqrt{-1}$. If $|z|^2 = 44200$, then $n$ is equal to _______
- Let $S = \{(m, n) : m, n \in \{1, 2, 3, \dots, 50\}\}$. If the number of elements $(m, n)$ in S such that $6^m + 9^n$ is a multiple of 5 is p and the number of elements $(m, n)$ in S such that $m + n$ is a square of a prime number is q, then $p + q$ is equal to ____________.
- Let $a_1 = 1$ and for $n \ge 1, a_{n+1} = \frac{1}{2} a_n + \frac{n^2 - 2n - 1}{n^2(n+1)^2}$. Then $\left| \sum_{n=1}^\infty \left( a_n - \frac{2}{n^2} \right) \right|$ is equal to ____________.
If $ \left(\frac{1}{^{15}C_0} + \frac{1}{^{15}C_1}\right)$ $\left(\frac{1}{^{15}C_1} + \frac{1}{^{15}C_2}\right)$ $\dots$ $ \left(\frac{1}{^{15}C_{12}} + \frac{1}{^{15}C_{13}}\right)$ = $\frac{\alpha^{13}}{^{14}C_0 ^{14}C_1 \dots ^{14}C_{12}} $, then $30\alpha$ is equal to __________.
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- Let $|A| = 6$, where A is a $3 \times 3$ matrix. If $|\text{adj}(3\text{adj}(A^2 \cdot \text{adj}(2A)))| = 2^m \cdot 3^n$, $m, n \in \mathbf{N}$, then $m + n$ is equal to ___________.
Important Questions from Algebra
Let S be the set of the first 11 natural numbers. Then the number of elements in $A = \{B \subseteq S : n(B) \geq 2$ and the product of all elements of B is even is ________.
- The number of $3 \times 2$ matrices A, which can be formed using the elements of the set $\{-2, -1, 0, 1, 2\}$ such that the sum of all the diagonal elements of $A^T A$ is 5, is ______
- The number of numbers greater than 5000, less than 9000 and divisible by 3, that can be formed using the digits 0, 1, 2, 5, 9, if the repetition of the digits is allowed, is ______
- Let $z = (1 + i)(1 + 2i)(1 + 3i) \dots (1 + ni)$, where $i = \sqrt{-1}$. If $|z|^2 = 44200$, then $n$ is equal to _______
- Let $S = \{(m, n) : m, n \in \{1, 2, 3, \dots, 50\}\}$. If the number of elements $(m, n)$ in S such that $6^m + 9^n$ is a multiple of 5 is p and the number of elements $(m, n)$ in S such that $m + n$ is a square of a prime number is q, then $p + q$ is equal to ____________.
