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 $l + m + n = 6 + 3 + 3 = 12$.
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