# How to negate predicates?

How can I go about negating predicates? It's asking me to shift a negation in as far inside the predicate as possible.

$$\forall x ((x \ge 100) \lor (x < 100))$$

I am quite new to discrete mathematics so would greatly appreciate a walkthrough. Thanks!

• $\neg \forall x.P(x) \iff \exists x.\neg P(x)$
– mrp
Aug 8, 2017 at 12:04

hint

$$\forall \to \;\;\exists$$ $$\ge \to \;\;<$$ $$\lor \to \;\;\land$$

so the negation is

$$\exists x \;: x <100 \;\; \land \;\; x\ge 100$$

remark

Your proposition is always true (tautology), thus its negation is always false (contradiction).

If you jut want the symbol which represents logical negation, you can visit Wikipedia's symbology page and see that you can use either

$$!$$

Or

$$¬$$