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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 $y<x$.

Femonto
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  • How about we start with you defining the properties in question: How do you define a relation for which trichotomy holds. What does it mean for a relation to be transitive? What does it mean if a relation is irreflexive? What does it mean for a relation to be asymmetrical? – amWhy Nov 04 '16 at 18:35
  • To put you on track: try out some relations on a set that has only $2$ elements. There are not so much ($2^4=16$ to be exact). – drhab Nov 04 '16 at 18:38
  • Thanks for your edit: More generally (and this gives you more options for finding an example): "A binary relation $R$ on $X$ is trichotomous if for all $x, y\in X,$ then exactly one of $xRy, yRx$ or $x=y$ holds. – amWhy Nov 04 '16 at 18:47

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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.

Jean Marie
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