Question
Let $A = \{ - 2, - 1,0,1,2,3,4\}$. Let R be a relation on A defined by xRy if and only if $2x + y \leq 2$. Let $l$ be the number of elements in R . Let m and n 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 :
Options
Solution
$R\{( - 2,a),( - 1,\text{ }b),(0,c),(1,\text{ }d),(2,e)\}$
$$\begin{matrix} & a = \{ - 2, - 1,0,1,2,3,4\};b = \{ - 2, - 1,0,1,2,3,4\} \\ & c = \{ - 2, - 1,0,1,2\};d = \{ - 2, - 1,0\} \\ & e = \{ - 2\} \end{matrix}$$
∴ No. of elements in R
$$= 7 + 7 + 5 + 3 + 1 = 23 = \mathcal{l} $$Minimum number of element to be added to make it reflexive $= m = 4 \Rightarrow \{(1,1),(2,2),(3,3),(4,4)\}$ minimum number of element to be added to make it symmetric $= n = 6$
for ' n '
$${\Rightarrow R = \{(3, - 2),(4, - 2),(2, - 1),(2,0),(3, - 1),(4, - 1)\} }{\therefore\mathcal{l} + m + n = 23 + 4 + 6 = 33}$$
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