Question

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_____.

Answer 1

Correct Answer: 7

The given relation \( R \) is: \[ R = \{(-4, 4), (-3, 3), (3, -2), (0, 1), (0, 0), (1, 1), (3, 3)\} \] For the relation to be reflexive, each element in \( A \) should be related to itself. The reflexive pairs that are missing are: \[ (-2, -2), (-4, -4), (-3, -3) \] Thus, we need to add these pairs to make the relation reflexive. For the relation to be symmetric, if \( (a, b) \) is in \( R \), then \( (b, a) \) must also be in \( R \). The missing symmetric pairs are: \[ (4, -4), (3, -3), (-2, 3), (1, 0) \] Thus, we need to add these pairs to make the relation symmetric. In total, the pairs we need to add are: \[ (-2, -2), (-4, -4), (-3, -3), (4, -4), (3, -3), (-2, 3), (1, 0) \] Thus, we need to add 7 elements to the relation.