Let A = $\{-2, -1, 0, 1, 2, 3\}$. Let R be a relation on A defined by xRy if and only if y = max{x,1}. Let $l$ be the number of elements in R. Let n and $m$ be the minimum number of elements required to be added in R to make it reflexive and symmetric relations respectively. Then $l + m + n$ is equal to
12
Relation R Elements (l)
The set is $A = \{-2, -1, 0, 1, 2, 3\}$. The relation $R$ is defined by $xRy$ if and only if $y = \text{max}\{x, 1\}$.
We find the pairs $(x, y)$ for each element $x \in A$:
- For $x = -2$, $y = \text{max}\{-2, 1\} = 1$. Pair: $(-2, 1)$.
- For $x = -1$, $y = \text{max}\{-1, 1\} = 1$. Pair: $(-1, 1)$.
- For $x = 0$, $y = \text{max}\{0, 1\} = 1$. Pair: $(0, 1)$.
- For $x = 1$, $y = \text{max}\{1, 1\} = 1$. Pair: $(1, 1)$.
- For $x = 2$, $y = \text{max}\{2, 1\} = 2$. Pair: $(2, 2)$.
- For $x = 3$, $y = \text{max}\{3, 1\} = 3$. Pair: $(3, 3)$.
Therefore, the relation $R = \{(-2, 1), (-1, 1), (0, 1), (1, 1), (2, 2), (3, 3)\}$.
The number of elements in $R$ is $l = |R| = 6$.
Reflexivity for R (n)
A relation $R$ on set $A$ is reflexive if $(x, x) \in R$ for all $x \in A$. We need the pairs $(-2, -2), (-1, -1), (0, 0), (1, 1), (2, 2), (3, 3)$ to be in $R$.
Checking against the existing $R = \{(-2, 1), (-1, 1), (0, 1), (1, 1), (2, 2), (3, 3)\}$:
- $(-2, -2)$ is missing.
- $(-1, -1)$ is missing.
- $(0, 0)$ is missing.
- $(1, 1)$ is present.
- $(2, 2)$ is present.
- $(3, 3)$ is present.
The minimum number of elements required to make $R$ reflexive is $n = 3$. These are $(-2, -2), (-1, -1), (0, 0)$.
Symmetry for R (m)
A relation $R$ is symmetric if $(x, y) \in R$ implies $(y, x) \in R$. We examine pairs $(x, y) \in R$ where $x \neq y$.
- $(-2, 1) \in R$. For symmetry, $(1, -2)$ must be in $R$. It is missing.
- $(-1, 1) \in R$. For symmetry, $(1, -1)$ must be in $R$. It is missing.
- $(0, 1) \in R$. For symmetry, $(1, 0)$ must be in $R$. It is missing.
- Pairs $(1, 1), (2, 2), (3, 3)$ satisfy symmetry trivially as $x=y$.
To make $R$ symmetric, we must add the pairs $(1, -2), (1, -1), (1, 0)$.
The minimum number of elements required to make $R$ symmetric is $m = 3$.
Sum l + m + n
We have calculated:
- $l = 6$ (number of elements in $R$)
- $n = 3$ (minimum elements for reflexivity)
- $m = 3$ (minimum elements for symmetry)
The required sum is $l + m + n = 6 + 3 + 3 = 12$.
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 __________.
- Three persons enter in a lift at the ground floor. The lift will go upto 10$^{\text{th}}$ floor. The number of ways, in which the three persons can exit the lift at three different floors, if the lift does not stop at first, second and third floors, is equal to _________.
- The number of 4-letter words, with or without meaning, which can be formed using the letters PQR PQRSTUVP, is ___________.
- 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 ____________.
