Question

Let S be the set of all points in a plane. Let R be a relation on S such that for any two points a and b, aRb iff b is within 1 cm from a. Then R is

• A. Equivalence relation
• B. Reflexive and symmetric not transitive
• C. Reflexive and transitive not symmetric
• D. Symmetric and transitive not reflexive

Answer 1

The correct option is B Reflexive and symmetric not transitive

R is reflexive since any point is at distance O from itself so that it is within 1 cm from itselt.
R is symmetric since if a point a is within 1 cm from another point b, then b is also within 1 cm from a.
R is not transitive. Let a, b, c, be three points in a straight lines in this order such that distance between a and b is $\frac{1}{2}$ cm and distance between band c is $\frac{2}{3}$ cm. Then aRb and bRc. But (a, c) distance between a $\text{/}\epsilon$ R since the and c is $(\frac{1}{2}+\frac{2}{3})=\frac{7}{6}cm>1cm.$