Question

Type of relations for aRb if and only if a<= b+1

Answer 1

Solution:

Consider relation R defined on integers (Z).

Reflexive:

$aRa\Rightarrow a\le a+1\Rightarrow True$

Thus, reflexive.

Symmetric:

$aRb\Rightarrow a\le b+1 \\ bRa \Rightarrow b\le a+1\Rightarrow False$

Thus, not symmetric.

Transitive:

$aRb\Rightarrow a\le b+1 \\bRc\Rightarrow b\le c+1$

$\Rightarrow a\le (c+1)+1=c+2\ or\ a\le c+1\Rightarrow aRc$

Thus, transitive.

Hence, given relation R is reflexive and transitive only for all $a,b,c\in Z$ .