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I am trying to understand what this equation actually means. I know that $\bigwedge$ means logical AND. So I can't really understand how it is being used in the context of a set of events $e \in \varepsilon$ here:

$$\bigwedge\limits_{e \in \varepsilon_u} \bigwedge\limits_{e' \in \varepsilon, e' \neq e} lb_e^{e'} \leq s_{e'} - s_{e} \leq ub_e^{e'}$$

where $\varepsilon$ is the set of all events, $\varepsilon_u$ is the subset of unexecuted events, $lb_e^{e'}$ is the lower bound time for the time between events $e$ and $e'$, $ub_e^{e'}$ is the upperbound, and $s_e$ and $s_{e'}$ are the times of the actual events $e$ and $e'$ respectively.

What are the two $\bigwedge$ signs in front actually meaning?

Thanks.

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2 Answers 2

up vote 1 down vote accepted

The symbol $\bigwedge$ is the allquantor (and $\bigvee$ the existential quantor). I think the majority of authors today prefer $\forall $ and $\exists$, respectively. Thus I would write your statement as $$\forall e\in\epsilon_u\colon \forall e'\in \epsilon\colon e'\ne e\to (lb_e^{e'}\le s_{e'}-s_e\le ub_e^{e'})$$

"For all $e\in \epsilon_u$ and all $e'\in\epsilon$ with $e'\ne e$, we have $lb_e^{e'}\le s_{e'}-s_e\le ub_e^{e'}$"

Remark: Of course the similarity of the symbols $\bigwedge$ and $\bigvee$ with $\land$ and $\lor$ is justified because $\bigwedge$ is to $\land$ what $\bigcup$ is to $\cup$ and $\Sigma$ is to $+$ and so on. I.e. $$\bigwedge_{n\in\mathbb N}\phi_n \iff \phi_1\land\phi_2\land\phi_3\land\ldots$$ $$\bigcup_{n\in\mathbb N}A_n = A_1\cup A_2\cup A_3\cup\ldots$$ $$\sum_{n\in\mathbb N}a_n = a_1+ a_2+a_3+\ldots$$

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The $\bigwedge$ symbol is not synonymous with the universal quantifier $\forall$ (in North America, at least). I believe it is being used as the logical AND in this case, although it can be refactored as you have done. –  ferson2020 Jan 30 '13 at 16:46
    
Thanks. I am more used to your other "For all" notation, and never encountered the allquanter, which is why I got confused. –  jbx Jan 30 '13 at 18:06

In words, this would mean for all pairs of events $(e, e')$ such that $e$ is unexecuted and $e'$ is distinct from $e$, the difference in time of $e$ from $e'$ is bounded by their lower and upper bounds.

The two $\bigwedge$ symbols just mean that all of the statements following, indexed over such pairs, are true simultaneously.

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Thanks for your explanation. Unfortunately I have to choose one answer and the other helped me understand a bit more, although yours is also OK once you understand it. So +1 for you too. –  jbx Jan 30 '13 at 18:09
    
Haha no problem, glad I could help! –  ferson2020 Jan 30 '13 at 18:16

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