z**** Define A quantifier, denoted

\[\begin{equation*} \exists x \>\> P(x), \end{equation*} \]

and read

“there exists *\(x\)* such that *\(P(x)\)*”

or

“for some *\(x\)*, *\(P(x)\)*”,

where *\(P\)* is the *quantified* predicate.

- Universal quantifier
Equivalent to the universal quantifier under negation by

\[\begin{align*} \exists x \> \lnot P(x) \> & \equiv \> \lnot \forall x \> P(x) \\ \lnot \exists x \> P(x) \> & \equiv \> \forall x \> \lnot P(x). \end{align*} \]

which is akin De Morgan’s laws, with

*\(\exists\)*generalizing disjunction.

There is no greatest integer *\(k\)* because

\[\begin{equation*} \forall k \in \mathbb{Z} \quad \exists \ell \in \mathbb{Z} \quad (\ell > k), \end{equation*} \]

that is

for every integer *\(k\)*,

there exists some integer *\(\ell\)*

greater than *\(k\)*,

such as *\(\ell = k + 1\)*.

© 2024 Rudolf Adamkovič under GNU General Public License version 3.

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