Let \(A=\{-3,-2,-1,0,1,2,3\}\). Let \(R\) be a relation on \(A\) defined by \(x R y\) if and only if \(0 \leq x^2+2 y \l…

\(A=\{-3,-2,-1,0,1,2,3\}, x R y \Longleftrightarrow 0 \leq x^2+2 y \leq 4, l=|R| \)

\(x= \pm 3, x^2=9: 0 \leq 9+2 y \leq 4 \Rightarrow-9 \leq 2 y \leq-5\)

\(\Rightarrow y=-3 \Rightarrow 2 \) pairs

\(x= \pm 2, x^2=4: 0 \leq 4+2 y \leq 4 \Rightarrow-4 \leq 2 y \leq 0\)

\(\Rightarrow y \in\{-2,-1,0\} \Rightarrow 6 \) pairs

\(x= \pm 1, x^2=1: 0 \leq 1+2 y \leq 4 \Rightarrow-1 \leq 2 y \leq 3\)

\(\Rightarrow y \in\{0,1\} \Rightarrow 4\) pairs

\(x=0, x^2=0: 0 \leq 2 y \leq 4\)

\(\Rightarrow y \in\{0,1,2\} \Rightarrow 3\) pairs

\( l=2+6+4+3=15\)

Reflexive on \(A\) means \((a, a) \in R \ \forall a \in A\), i.e. \(0 \leq a^2+2 a \leq 4\)

\(a=-3,-2,0,1\) satisfy it; \(a=-1,2,3\) do not satisfy it \(\Rightarrow m=3\)

\(l+m=15+3=18\)