two examples of relations with some conditions (transitive, irreflexive, asymmetric)

Question

I need help with this two exmples, I have no idea.

a example of a relation holds trichotomy and that it is not transitive.

a example of a relation that is irreflexive and that it is not asymmetrical.

R holds trichotomy if $x<y$ or $x=y$ or $y<x$.

Answer 1

First counter-example: Consider relation R whose boolean representation is:

$$\pmatrix{1&1&0\\0&1&1\\1&0&1}$$

i.e. xRx, yRy, zRz, xRy, yRz and zRx are true and only them.

• Clearly, R is trichotomous,

• whereas R is not transitive because we have xRy and yRz without having xRz.