Let A= {- 2, - 1, 0, 1, 2, 3, 4}. Let R be a relation on A defined by xRy if and only if $$|2x + y| \leq 3$$. Let l be the number of elements in R. Let m and n be the minimun number of elements required to be added in R to make it reflexive and symmetric relations respectively. Then l+ m + n is equal to:

$$A=-2,-1,0,1,2,3,4,\quad xRy\Leftrightarrow|2x+y|\le3$$

Count pairs for each (x):

• $$(x=-2:\ y=1,2,3,4\Rightarrow4)$$
• $$(x=-1:\ y=-1,0,1,2,3,4\Rightarrow6)$$
• $$(x=0:\ y=-2,-1,0,1,2,3\Rightarrow6)$$
• $$(x=1:\ y=-2,-1,0,1\Rightarrow4)$$
• $$(x=2:\ y=-2,-1\Rightarrow2)$$

l=4+6+6+4+2=22

For reflexive relation, missing:
(-2,-2),(2,2),(3,3),(4,4)
Hence m=4

For symmetry, missing reverse pairs:
(-1,-2),(0,-2),(0,2),(1,-1),(1,2),(2,-2),(2,0)
Hence n=7
l+m+n=22+4+7=33