1

Example of a binary relation that is transitive and not negatively transitive:

My try: $1\neq 2$ and $2\neq 1$ does not imply $1\neq 1$ Not neg transitive.

But if $1=2$ and $2=1$ then $1=1$ by transitivity.

Example of a binary relation that is negatively transitive but not transitive.

My try: Need help on this.

OGC
  • 2,305

3 Answers3

1

I think the following would be a good example:

Let X = {x,y,z} and the binary relation on X, R = {(x,y)} (that is, xRy),

This is transitive, since only two elements are related. However, it is NOT negatively transitive because $\neg$zRy and $\neg$xRz but xRy!

Arturo
  • 11
1

Hint: What if you just take $\neq$ as your relation? Why is this negatively transitive?

Ben Grossmann
  • 225,327
  • Oh I see. I made a mistake in changing the relation in one example. – OGC Jan 21 '14 at 01:10
  • You haven't made any mistakes as far as I can tell. The point is that the relation defined by "$=$" is transitive, but not negatively transitive. Likewise, the relation $\neq$ is negatively transitive, but not transitive. – Ben Grossmann Jan 21 '14 at 01:12
  • So I just do the opposite for transitive and neg transitive example? Just switch the lines from example 1? – OGC Jan 21 '14 at 01:15
  • Pretty much. By that first line, $\neq$ is not transitive. By that second line, $\neq$ is negatively transitive. – Ben Grossmann Jan 21 '14 at 01:34
0

"Dictionary letters" order is one. Consider the following set with $5$ letters, say:

$$\{(a,c), (a,d), (a,e), (b,d), (b,e), (c,e)\}$$

In order for the coordinates to be in the set, the letter must come in earlier before any letter in the alphabet.

NasuSama
  • 3,364