Let A={1,2,3,4,5}. Let R be a relation on A defined by x R y if and only if 4 x ≤ 5 y. Let m be the number of elements in R and n be the minimum number of elements from A × A that are required to be added to R to make it a symmetric relation. Then m + n is equal to :

Solution

Maths sir · 4 steps

List R row by row for each x, then add a mirror pair for each pair whose mirror is missing.

1. Rewrite the condition. 4x ≤ 5y means y ≥ 4x/5. For each x in A, keep every y in A that is at least 4x/5.

   x	y ≥ 4x/5	Allowed y	Pairs in R	Mirror missing
   1	y ≥ 0.8	1, 2, 3, 4, 5	5	4: (1,2),(1,3),(1,4),(1,5)
   2	y ≥ 1.6	2, 3, 4, 5	4	3: (2,3),(2,4),(2,5)
   3	y ≥ 2.4	3, 4, 5	3	2: (3,4),(3,5)
   4	y ≥ 3.2	4, 5	2	0: (5,4) is already in R
   5	y ≥ 4	4, 5	2	0: (4,5) is already in R

2. Count m. Adding the fourth column gives m = 5 + 4 + 3 + 2 + 2 = 16.
3. Count n. A diagonal pair such as (3,3) is its own mirror. Of the 11 off-diagonal pairs, (4,5) and (5,4) are mirrors of each other and are both in R. Each of the other 9 needs its mirror added, so n = 9.
4. So. m + n = 16 + 9 = 25, which is option (c).

The idea to keep. To make a relation symmetric, add (b,a) for every (a,b) in R whose mirror is missing. Diagonal pairs and pairs already present with their mirrors need nothing.

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