Let $A = \{-3, -2, -1, 0, 1, 2, 3\}$ and $R$ be a relation on $A$ defined by $xRy$ if and only if $2x - y \in \{0, 1\}$. Let $\ell$ be the number of elements in $R$. Let $m$ and $n$ be the minimum number of elements required to be added to $R$ to make it reflexive and symmetric relations, respectively. Then $\ell + m + n$ is equal to:
Step-by-Step Solution
Key Concept: Translate the finite relation rule into explicit admissible ordered pairs and count the required set.
For the relation $xRy$ where $2x - y \in \{0, 1\}$:
**Case 1: $2x - y = 0$ (i.e., $y = 2x$)**
Pairs: $(0, 0)$, $(-1, -2)$, $(1, 2)$
**Case 2: $2x - y = 1$ (i.e., $y = 2x - 1$)**
Pairs: $(0, -1)$, $(1, 1)$, $(2, 3)$, $(-1, -3)$
**Total pairs in $R$:**
$\ell = 7$ pairs: $(0, 0)$, $(-1, -2)$, $(1, 2)$, $(0, -1)$, $(1, 1)$, $(2, 3)$, $(-1, -3)$
**For Reflexivity:**
Required pairs: $(x, x)$ for all $x \in A$
Already in $R$: $(0, 0)$, $(1, 1)$
Need to add: $(-3, -3)$, $(-2, -2)$, $(-1, -1)$, $(2, 2)$, $(3, 3)$
Thus $m = 5$
**For Symmetry:**
For each $(x, y) \in R$, we need $(y, x) \in R$
Pairs needing symmetric counterparts: $(-1, -2)$, $(1, 2)$, $(0, -1)$, $(2, 3)$, $(-1, -3)$
Required symmetric pairs: $(-2, -1)$, $(2, 1)$, $(-1, 0)$, $(3, 2)$, $(-3, -1)$
Thus $n = 5$
**Final Answer:**
$\ell + m + n = 7 + 5 + 5 = 17$
Correct Answer: 2