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 $l+m+n = 7 + 5 + 5 = 17$.
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