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, a R b iff b is within 1 centimetre from a. Check R for reflexivity, symmetry and transitivity.

Answer 1

$aRb\iff \text{b is within 1 cm from a}$
Reflexivity:
R is reflexive since any point is at distance 0 from itself so that it is within 1 centimetre from itself.
Symmetry:
R is symmetric since if a point a is within 1 centimetre from another point b, then b is also within 1 centimetre from a.
Transitivity:
Let a,b,c be three points in a straight line in this order such that distance between a and b is $\frac{1}{2}$ centimetre and distance between b and c is $\frac{3}{4}$ centimetre .
Then a R b and b R c.
But $a(\sim R)$ c since the distance between a and c is $(\frac{1}{2}+\frac{3}{4})=\frac{5}{4}$ centimetre which is greater than 1 centimetre.
Hence, R is not transitive.