Calculating Relation R Elements (l)
The set is $A = \{-3, -2, -1, 0, 1, 2, 3\}$. A relation $R$ on $A$ is defined by $xRy$ if and only if $2x-y \in \{0, 1\}$. This means $y = 2x$ or $y = 2x-1$. We list the pairs $(x, y)$ in $A \times A$ that satisfy this condition:
- For $x = -1$: $y = 2(-1) = -2$ or $y = 2(-1) - 1 = -3$. Pairs: $(-1, -2), (-1, -3)$.
- For $x = 0$: $y = 2(0) = 0$ or $y = 2(0) - 1 = -1$. Pairs: $(0, 0), (0, -1)$.
- For $x = 1$: $y = 2(1) = 2$ or $y = 2(1) - 1 = 1$. Pairs: $(1, 2), (1, 1)$.
- For $x = 2$: $y = 2(2) = 4$ (not in $A$); $y = 2(2) - 1 = 3$. Pair: $(2, 3)$.
Other values of $x$ in $A$ do not yield $y$ values within $A$. Thus, the relation $R$ is:
$R = \{(-1, -2), (-1, -3), (0, 0), (0, -1), (1, 2), (1, 1), (2, 3)\}$
The number of elements in $R$ is $l = 7$.
Calculating Minimum Elements for Reflexivity (m)
A relation $R$ is reflexive if $(x, x) \in R$ for all $x \in A$. The condition for $(x, x) \in R$ is $2x - x \in \{0, 1\}$, which simplifies to $x \in \{0, 1\}$.
The elements $x \in A$ for which $(x, x)$ must be in $R$ are $\{-3, -2, -1, 0, 1, 2, 3\}$.
The pairs $(x, x)$ already present in $R$ are $(0, 0)$ and $(1, 1)$.
The pairs that need to be added to make $R$ reflexive are those $(x, x)$ for $x \in A \setminus \{0, 1\}$: $\{-3, -2, -1, 2, 3\}$.
These pairs are: $(-3, -3), (-2, -2), (-1, -1), (2, 2), (3, 3)$.
The minimum number of elements to add for reflexivity is $m = 5$.
Calculating Minimum Elements for Symmetry (n)
A relation $R$ is symmetric if whenever $(x, y) \in R$, then $(y, x) \in R$. We examine the pairs in $R$ and check if their reverse pairs exist in $R$.
- $(-1, -2) \in R$. Is $(-2, -1) \in R$? Check: $2(-2) - (-1) = -4 + 1 = -3 \notin \{0, 1\}$. Need to add $(-2, -1)$.
- $(-1, -3) \in R$. Is $(-3, -1) \in R$? Check: $2(-3) - (-1) = -6 + 1 = -5 \notin \{0, 1\}$. Need to add $(-3, -1)$.
- $(0, 0) \in R$. $(0, 0)$ is symmetric with itself.
- $(0, -1) \in R$. Is $(-1, 0) \in R$? Check: $2(-1) - 0 = -2 \notin \{0, 1\}$. Need to add $(-1, 0)$.
- $(1, 2) \in R$. Is $(2, 1) \in R$? Check: $2(2) - 1 = 4 - 1 = 3 \notin \{0, 1\}$. Need to add $(2, 1)$.
- $(1, 1) \in R$. $(1, 1)$ is symmetric with itself.
- $(2, 3) \in R$. Is $(3, 2) \in R$? Check: $2(3) - 2 = 6 - 2 = 4 \notin \{0, 1\}$. Need to add $(3, 2)$.
The pairs that need to be added for symmetry are: $\{(-2, -1), (-3, -1), (-1, 0), (2, 1), (3, 2)\}$.
The minimum number of elements to add for symmetry is $n = 5$.
Final Calculation (l + m + n)
We have $l=7$, $m=5$, and $n=5$.
The required sum is $l+m+n = 7 + 5 + 5 = 17$.
Similar Questions
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Important Questions from Algebra
- Let $\alpha = \frac{-1 + i\sqrt{3}}{2}$ and $\beta = \frac{-1 - i\sqrt{3}}{2}$, $i = \sqrt{-1}$. If $(7 - 7\alpha + 9\beta)^{20} + (9 + 7\alpha - 7\beta)^{20} + (-7 + 9\alpha + 7\beta)^{20} + (14 + 7\alpha + 7\beta)^{20} = m^{10}$, then $m$ is _________
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 ______
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