Let A={-4,-3,-2,0,1,3,4} and R={(a, b) ∈ A × A: b=|a| or b²=a+1} be a relation on A. Then the minimum number of elements, that must be added to the relation R so that it becomes reflexive and symmetric, is __________

Solution

Maths sir · 3 steps

First list R pair by pair, then add the missing diagonal pairs and the missing reverse pairs.

1. a	From b = |a|	From b² = a + 1
   -4	(-4, 4)	b² = -3, none
   -3	(-3, 3)	b² = -2, none
   -2	b = 2 ∉ A, none	b² = -1, none
   0	(0, 0)	b = ± 1, only (0, 1)
   1	(1, 1)	b² = 2, none
   3	(3, 3)	b = ± 2, only (3, -2)
   4	(4, 4)	b² = 5, none

2. Make it reflexive. Every (a, a) must be in R. We already have (0,0), (1,1), (3,3) and (4,4). Add (-4,-4), (-3,-3) and (-2,-2): 3 pairs.
3. Make it symmetric. The pairs in R that are not on the diagonal are (-4,4), (-3,3), (0,1) and (3,-2). None of their reverses is in R, so add (4,-4), (3,-3), (1,0) and (-2,3): 4 pairs.
4. So. The minimum number of pairs to add is 3 + 4 = 7.

The common slip: 6. Taking only b = 2 from b² = 4 misses (3,-2) and its reverse (-2,3), which gives 6. The root b = -2 is in A, so it counts.

The idea to keep. Write the relation out in full before testing any property. For reflexive, count the missing (a,a) pairs. For symmetric, count each off-diagonal pair whose reverse is missing.

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