Properties of $(m, n) ∈ R \iff m > n$

Asked Mar 30 '24 at 18:26

Active Mar 30 '24 at 20:17

Viewed 42 times

Given the following relation, which properties does it fulfill?

Is my reasoning correct or am I missing something?

$(m, n) ∈ R \iff m > n$

1.) reflexivity

no, $1 > 1$ (trivially false)

2.) symmetry

no, since $x > y\not\rightarrow y > x$, since $2 > 3 \not\rightarrow 3 > 2$

3.) antisymmetry

no, since $(2 > 2)\wedge(2>2) \not\rightarrow 2 = 2 $ (trivially false)

4.) transitivity

yes, since one can prove that $\forall m,n,r\in R:((m> n ) \wedge (n>r)) \rightarrow m > r$

edited Mar 30 '24 at 20:17

md2perpe

asked Mar 30 '24 at 18:26

Antisymmetry is actually true. – Arthur Mar 30 '24 at 18:29
I had that suspicion. Why is that so? Because actually, you can't find a counter-example, right? But how would one prove that? – einzigartigerhummer Mar 30 '24 at 18:29
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Antisymmetry is incorrect. To show a relation is not antisymmetric, you'd need to find $x,y$ such that $x<y$ and $y<x$ but $x \neq y$. Can you find such $x,y$ here? – Sambo Mar 30 '24 at 18:30
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Antisymmetry in this case expresses that if a is larger than b, and b is larger than a, then a = b. Is this true? In your written example, it is trivially true because 2 > 2 is false, and so the implication vacuously holds – Michael Carey Mar 30 '24 at 18:30
Could I argue that $((x > y) \wedge (y > x))$ is always vacuously false, hence $x = y$ is also vacuously false? This is an implication, where $F$ (false) $\implies F$ (false), hence its truth value is $T$ (true)? – einzigartigerhummer Mar 30 '24 at 18:36
Related: The relation "is strictly higher than" is considered antisymmetric? – peterwhy Mar 30 '24 at 18:38
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Yes, that's the right idea. But, you cannot know that x = y is false from the false premise. However you do get that the premise in the implication is false, which means the implication is true- regardless of if the conclusion is true or false – Michael Carey Mar 30 '24 at 18:41
Now I understand, thank you very much! :) – einzigartigerhummer Mar 30 '24 at 18:50
Use \iff to get $\iff$. – md2perpe Mar 30 '24 at 20:18

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