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Suppose this state for the relation schema Company

State Company Relation Schema

I'm stuck trying to understand how, why and when I should use Quantifiers (Existencial and Universal)

Examples:

Example 1

Here I understand how the existencial is working, all the formula bounded to him would evaluate to TRUE, if at least one of all the possible tuples that d can take evaluates to TRUE (please tell me if this reasoning is okay)

Example 2

Here is where things get messy, I don't understand what's actually doing that Universal Quantifier in NOT(PROJECT(x)), and why is it using x as a tuple in PROJECT in this part NOT(x.Dnum = 5), how we know that x is in PROJECT ? Is something about the first part with the Universal Quantifier? And why the Universal Quantifier is just bounding NOT(PROJECT(x))?

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