Let ε = 0.0005 and R_ε be the relation {(x, y) ∈ ℝ² : |x − y| < ε}, interpreted as 'approximately equal'. R_ε is (A) reflexive (B) symmetric (C) transitive.

1. A(A) and (B) only true
2. B(B) and (C) only true
3. C(A) and (C) only true
4. D(A), (B) and (C) true

Pick an option to check yourself, or open the answer below.

Show answer & explanation

Correct answer: D. (A), (B) and (C) true

Explanation

|x − x| = 0 < ε, so R_ε is reflexive, and |x − y| = |y − x| makes it symmetric. It is not transitive, so the correct answer is (A) and (B) only.

Note on the official key: the official key says option A. Official/PW key is D. R_ε is not transitive: 0 and 0.0004 are related, 0.0004 and 0.0008 are related, but 0 and 0.0008 are not.