Find these relations on $\mathbb{N}$.

Question

Give an example of a relation on $\mathbb{N}$ which is,

1. reflexive, transitive but not symmetric
2. transitive, symmetric but not reflexive
3. reflexive, symmetric but not transitive
4. anti-symmetric, transitive but not reflexive.

I found relations for 1, 2 :

1. $R = \{(x,y) | x\leq y\;and\; x,y \in \mathbb{N} \}$
2. $R = \{(x,y) | x,y\;are\;both \;even\;and\; x,y \in \mathbb{N} \}$ \

I am stucked at finding relations on $\mathbb{N}$ for 3, 4. Please help.

Answer 1

There are many (in fact, uncountably infinitely many) examples for these. Here are a few real life examples that came quickly to my mind that you can formalize and translate to your own scenario:

For (3)

• Living within a mile of one another. I might live three quarters of a mile from my friend. My friend then clearly lives the same distance away from me. I live within a mile of myself... Now, my friend and I have another friend who lives in the same direction but further away. Although that other friend lives within a mile of my first friend, he happens to live over a mile away from me.

$|x-y|<k$

• Being related by blood (within 4 generations, just so you can't argue that we're all related to Adam and Eve in some way). I am related to myself. I am related to both of my parents. My parents are related to me. My parents are not related to each other.

$\gcd(x,y)>1$ or $x=1$ or $y=1$

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For (4)

• Nothing is related to anything

The empty relation is still a valid example of a relation. It may be a trivial relation with rather uninteresting properties that is rarely if ever used except as an example or counterexample but there is no denying its existence. Make sure you fully understand why the empty relation satisfies the desired properties.

• Consider your answer for (1) and tweak it slightly to make it no longer reflexive.

Replace $\leq$ with $<$. That removes reflexivity.