Exclude the tuples of no interest in relational calculus

Question

What is the meaning of the following expression

(∀x)(¬P(x) OR ¬(x.Dnum=5))

As far as I know, it means for all x, P(x) is false, but in the book where I read this, it states that all tuples which are not in P.

1. Is this statement right or equivalent to what i think? Please explain.

The author also says that in order to exclude from universal classification all tuples that we are not interested in, is by making the condition TRUE for all such tuples.

2. What does it means and how can we exclude tuples like that?

3. Also please tell that whether AND means INTERSECTION and OR means UNION in relational calculus?

Answer 1

If I interpret correctly the syntax, the formula (I've used $\lor$ for OR) :

$(¬P(x) \ \lor \ ¬(x.Dnum=5))$

will be satisfied (will be TRUE) by a tuple $x$ when :

either $x$ is not $P$'s or $x$ does not have "lenght" $5$.

If so, the formula :

$(∀x)(¬P(x) \ \lor \ ¬(x.Dnum=5))$

will be TRUE if there are no tuple that are $P$s and have lenght $5$.

Note : the formula is equivalent to: $(¬∃x)(P(x) \land x.Dnum=5)$.