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Suppose there is a tournament involving $b$ teams where $b \ge 2$. It is viable to name the teams as $T_1, T_2, \cdots T_b$ so that $T_r$ beats $T_{r+1}$ for any $r$ from $1$ to $b-1$.

Here is what I've done

$\forall r \in \{1,\cdots, b-1\}$, $T_r$ beats $T_{r+1}\iff b \ge 2$

Is this the correct way to write the symbolic form of the above statement?

Don Metro
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  • I have two questions: Are there $b$ teams or $n$ teams? (I guess $b$). Why is the claim that any team on the list beats the next team on the list equivalent to $b\geq 2$? – Mankind Feb 07 '19 at 21:17
  • Possible duplicate of Prove that every tournament contains at least one Hamiltonian path. – Mike Earnest Feb 07 '19 at 21:21
  • yes @Mankind it's supposed to be b what I'm trying to say in symbolic form is that for all r from 1 to b-1 Tr beats Tr+1 iff b is more than 2. b is more than or equal to two is because to have a match we need 2 or more teams. From the textbook it says it's a round robin theory – Don Metro Feb 07 '19 at 22:27
  • thank you for the link @MikeEarnest. After reading the link I'm still at lost though, is the symbolic for of the statement I wrote above is correct or on the right path? – Don Metro Feb 07 '19 at 22:36
  • Not really, no. The statement in English is: "For all $b\ge 2$ and for all tournaments with $b$ teams there exists a list $T_1,\dots,T_b$ such that for all $r\in {1,2,\dots,b-1}$, $T_r$ beats $T_{r+1}$. So you are missing several $\forall$'s and $\exists$'s, and the $\iff$ is not right. – Mike Earnest Feb 07 '19 at 22:41
  • thank you for the reply @MikeEarnest. I'm curious though, why is the ⟺ isn't right? Isn't the ⟺ means if and only if? why is it not suitable to use it for this statement? :D – Don Metro Feb 07 '19 at 23:15
  • No, $\iff$ is not right. If and only if would mean that $b<2$ implies something, but we are not saying anything in the case $b<2$. – Mike Earnest Feb 08 '19 at 01:48

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