Find relations on the real number: transitive and/or antisymmetric

Question

$$I\ am\ searching\ for\ a\ relation\ on\ the\ real numbers\ (\mathbb R ),\ which\ sould\ be:$$

1. antisymmetric and transitive
2. antisymmetric and NOT transitive
3. NOT antisymmetric ,but transitive
4. NOT antisymmetric and NOT transitive

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My ideas:

1. antisymmetric and transitive : $$ the "\leq" relation $$ $$ antisym: x\leq y \wedge y\leq x \Rightarrow x=y$$ $$ trans: x\leq y\wedge y\leq z \Rightarrow x\leq z $$

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2. antisymmetric and NOT transitive: $$\neq relation$$

   $$antisym: x\neq y \wedge y\neq x\Rightarrow x\neq y $$ $$ trans: Example: 2\neq 3 \wedge 3\neq 2 \Rightarrow 2\neq 2 => contradiction!$$

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3. NOT antisymmetric ,but transitive: $$ "<" relation $$

$$ antisym: x<y \wedge y<x \Rightarrow that's\ always\ wrong!$$ $$ trans: x<y \wedge y<z \Rightarrow x<z$$

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4. NOT antisymmetric and NOT transitive: $$ xRy \Leftrightarrow |x-y|=1$$

$$ antisym: 0R1 \wedge 1R0 $$ $$ trans: 0R1 \wedge 1R0 \Rightarrow 0R2 \ \ contradiction!$$

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Questions: Are my relations right? I think there is a mistake in the antisymmetric case of relation 3?

Answer 1

1) Correct.

2) $\neq$ is not antisymmetric (and not transitive, so it can be used for 4). For that $[x\neq y\wedge y\neq x]\Rightarrow x=y$ must be true, and that is not the case.

3) $<$ is antisymmetric. This because $\neg[x<y\wedge y<x]$ is true and consequently $\neg[x<y\wedge y<x]\vee x=y$ (equivalent with: $[x<y\wedge y<x]\Rightarrow x=y$) is true.

4) Correct.