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Existential and Universal quantifiers in DBMS

I understand what they are and I have seen examples regarding these. One such example is

{t.Fname,t.Lname|Employee(t) AND (∃d)(Department(d) AND d.Dname='Research' AND d.Dnumber=t.Dno)}

Now what is the difference between above and this

{t.Fname,t.Lname|Employee(t) AND Department(d) AND d.Dname='Research' AND d.Dnumber=t.Dno}

And how is

(∀x) (P(x)) ≡ NOT (∃x) (NOT (P(x)))

Can someone please explain ?

Solution

  • For every x P(x) means that all x satisfy P, which means that there doesn't exist an x which doesn't satisfy P, hence

    (∀x) (P(x)) ≡ NOT (∃x) (NOT (P(x)))