Let and Let be the relation \(\left\{(x, \in R^2:|x-y|< \in
Duration: 4 min
2022
Let ε = 0.0005 and let Re be the relation
Re = { (x, y) ∈ R² : |x − y| < ε }
Re could be interpreted as the relation "approximately equal".
Re is:
(A) Reflexive
(B) Symmetric
(C) Transitive
Choose the correct answer from the options given below.
- A.
(A) and (B) only true
- B.
(B) and (C) only true
- C.
(A) and (C) only true
- D.
(A), (B) and (C) true
Attempted by 304 students.
Show answer & explanation
Correct answer: A
Consider the relation R_e = {(x,y) in R^2 : |x - y| < 0.0005}.
Reflexive: For any real x, |x - x| = 0, and 0 < 0.0005, so (x,x) belongs to R_e. Thus the relation is reflexive.
Symmetric: If |x - y| < 0.0005 then |y - x| = |x - y| < 0.0005, so (y,x) is also in R_e. Therefore the relation is symmetric.
Transitive: The relation is not transitive. For a counterexample, take x = 0, y = 0.00025, and z = 0.0005. Then |x - y| = 0.00025 < 0.0005 and |y - z| = 0.00025 < 0.0005, but |x - z| = 0.0005 which is not < 0.0005, so (x,z) is not in R_e. Hence transitivity fails.
Conclusion: The relation is reflexive and symmetric only.
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