Let A={-2,-1,0,1,2,3}. Let R be a relation on A defined by x R y if and only if y=max {x, 1}. 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

Solution

Maths sir · 4 steps

List all six pairs first, then count what is missing for reflexive and for symmetric.

1. Build R. Each x in A gives exactly one y = max{x, 1}. Every x ≤ 1 goes to 1, and x = 2, 3 go to themselves.

   x	y = max{x, 1}	Pair in R	Diagonal (x, x) present?
   -2	1	(-2, 1)	no
   -1	1	(-1, 1)	no
   0	1	(0, 1)	no
   1	1	(1, 1)	yes
   2	2	(2, 2)	yes
   3	3	(3, 3)	yes

2. Count l and m. R has 6 pairs, so l = 6. For reflexive, every (x, x) must be present. (-2,-2), (-1,-1) and (0,0) are missing, so m = 3.
3. Count n. For symmetric, every (a, b) needs its reverse (b, a). The pairs (1,1), (2,2) and (3,3) are their own reverses. The other three need (1,-2), (1,-1) and (1,0), so n = 3.
4. So. l + m + n = 6 + 3 + 3 = 12, which is option (b).

The idea to keep. For the fewest additions: reflexive needs only the missing (x, x) pairs, and symmetric needs only the reverse of each non-diagonal pair. Diagonal pairs are already symmetric.

Answer from the NTA final answer key. Solution by PrepWiser.