Which of the following represent the reflexive transitive closure of
Which of the following represent the reflexive transitive closure of R={x,y | y=x+1 and x,y∈N}
- A.
R={x,y | y>x and x,y∈N}
- B.
R={x,y | y=x+1 or y=x and x,y∈N}
- C.
R={x,y | yx and x,y∈N}
- D.
R={x,y | y ≥ x and x,y∈N}
Attempted by 51 students.
Show answer & explanation
Correct answer: D
Given relation: R = {(x,y) ∈ N×N | y = x + 1}.
Transitive closure (without reflexivity): Applying the relation repeatedly gives pairs of the form y = x + n for any integer n ≥ 1.
n = 1: y = x + 1 (original relation)
n = 2: y = x + 2 (composition of the relation with itself)
In general: y = x + n for all integers n ≥ 1
Reflexive transitive closure: Include the case n = 0 (reflexivity), so y = x + n for n ≥ 0. This condition is equivalent to y ≥ x.
Conclusion: The reflexive transitive closure is {(x,y) ∈ N×N | y ≥ x}.