Properties of the relation $(x,y) \in R$ if $| x-y | = 2 $

Question

Is the following relation reflexive, symmetric, transitive, anti-symmetric and/or partial order :

$$(x,y) \in R \text{ if }| x-y | = 2 $$

I think it's reflexive, I don't understand how to find for the other ones.

Answer 1

A relationship $R$ is reflexive if, for all $x$, it's true that $x R x$.

$$\textsf{Is }\lvert x-x\rvert = 2 \textsf{ universally true?}$$

Clearly no.

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A relationship $R$ is reflexive if, for all $x,y$, it's true that $x R y\to y R x$

$$\textsf{Is }\lvert y-x\rvert = 2 \textsf{ true whenever }\lvert x-y\rvert = 2\textsf{ is true?}$$

Obviously, yes.

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A relationship $R$ is transitive if, for all $x,y,z$, it's true that $(x R y\wedge y R z)\to x R z$

$$\textsf{Is }\lvert x-z\rvert = 2 \textsf{ true whenever }\lvert x-y\rvert = 2\textsf{ and }\lvert y-z\rvert = 2\textsf{ are both true?}$$

Not telling. Is it?

A relationship $R$ is antisymmetric if, for all $x,y$, it's true that $(x R y\wedge y R x)\to x = y$

$$\textsf{Is }x=y \textsf{ true whenever }\lvert x-y\rvert = 2\textsf{ and }\lvert y-x\rvert = 2\textsf{ are both true?}$$

Not telling. Is it?

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The way to falsify each of these claims is to find a counterexample.   Only by verifying a counterexample is not possible can you state that a claim is true.

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A relationship $R$ is a partial order relation if it is reflexive, antisymetric, and transitive .

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