This is actually proof problem A relation R is defined on Z

This is actually proof problem.

A relation R is defined on Z as follows: For all a,b Z, aR b if and only
if |a - b| 3. Is relation R an equivalence relation on R? If not, is R reflexive,
symmetric, or transitive? Justify all conclusions.

Solution

Answer :

A relation R is defined on Z as follows: For all a,b Z, aR b if and only
if |a - b| 3.

Then R is reflexive since for any a Z, we have |a - a| 3.

R is symmetric since  for all a,b Z, if |a - b| 3. then |b - a| 3.

R is not transitive since for all a,b ,c Z, if |a - b| 3 and  |b - c| 3 then

|a - c| = |a - b+ b - c| |a - b | +| b - c| 3 + 3 = 6 hence | a - c | 6.

As R is not transitive implies that R is not an equivalence relation