Formal mathematical statements are often built by predicates.

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**Definition 1** (Predicate). *If \(X\) is any set and \(A(x)\) is a logical statement depending on
\(x \in X\) (and true or false for
every \(x\in X\)), we call \(A(x)\) a **predicate* with variable
\(x\). Usually, one writes simply \(A(x)\) instead of \(A(x)=\) true.

**Example 2**. \[X=\mathbb{R}\qquad A(x) \;=\; x < 0\]
Then we can define the set \[\{ x \in X :
A(x) \} = \{ x \in \mathbb{R} : x < 0 \}\]

**Definition 3** (Quantifiers \(\forall\) and \(\exists\)). *We use \(\forall\) (“for all”) and \(\exists\) (“it exists”) and call them
**quantifiers*. Moreover, we use the double point “ \(:\) ” inside the set brackets, which means
“that fulfil”.

The quantifiers and predicates are very useful for a compact
notation:

\(\forall x \in X : A(x)~~\)
*for all \(x\in X\) \(A(x)\) is true*

\(\exists x \in X : A(x)~~\)
*there exists at least one \(x\in
X\) for which \(A(x)\) is
true*

\(\exists! x \in X : A(x)~~\)
*there exists exactly one \(x\in X\)
for which \(A(x)\) is
true*

Negation of statements with quantifiers:

**Example 4**. There is no greatest natural number:

\(A(n) \;=\; n\) is the greatest
natural number

In our notation: \(\neg (\exists n \in
\mathbb{N} : A(n))\) this is the same as \(\forall n \in \mathbb{N} : \neg A(n)\),
i.e. *Each \(n \in \mathbb{N}\) is
not the greatest natural number* . But this is clear, because \(n+1 > n\).

**Rule of Thumb 5** (Negation of the quantifier (\(\forall\) and \(\exists\))).

\(\neg\forall = \exists \neg\) and
\(\neg\exists = \forall \neg\)

**Example 6**. The set \(M:=\{x\in\mathbb{Z} \colon x^2=25\}\) is
defined by the set of each integer \(x\) that squares to 25. We immediately see
that this is just \(-5\) and \(5\).

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