# Membership and inclusions

## Membership (possessive relations)

If \(x\) is an element in a set \(A\) we write \[ x \in A \quad \text{ or }\quad A \ni x, \] and if not \[ x \notin A \quad\text{ or }\quad A \not\ni x. \]

**Ex.**

- The
**rationals**can be constructed from elements in \(\mathbb Z\) and \(\mathbb N\):

\[ \mathbb Q = \{ a/b \colon a \in \mathbb Z, b \in \mathbb N\} \]

- \(\sqrt{2}\) is a real number, but not a rational one:

\[ \sqrt{2} \in \mathbb{R}, \qquad \sqrt{2} \notin \mathbb{Q}. \]

## Quantifiers

Quantifiers are used to abbreviate notation. The most important ones are:

- \( \forall \quad \) Universal quantifier: 'For any','for all'
- \( \exists \quad \) Existential quantifier: 'There exists'
- \( ! \quad \) Uniqueness quantifier: 'a unique'

**Ex.**For any real number \(x\) there exists a unique real number \(-x\) with the property that the sum of \(x\) and \(-x\) is zero: \[ \forall\: x \in \mathbb{R} \qquad \exists ! \:\; (-x) \in \mathbb{R}; \qquad x + (-x) = 0.\]

## Inclusions

A set \(A\) is a **subset** of a set \(B\) if any element in \(A\) is also an element in \(B\):
\[
A \subset B \quad (\text{or } A \subseteq B) \quad\stackrel{\text{def.}}{\Longleftrightarrow}\quad [x \in A \Rightarrow x \in B]
\]
A subset \(A \subset B\) can also be a **proper subset** of \(B\):
\[
A \subsetneq B \quad\stackrel{\text{def.}}{\Longleftrightarrow}\quad A \subset B \text{ but } A \neq B.
\]

**Ex.**

- The natural numbers is a subset of the set of non-negative integers, which is a proper subset of the set of real numbers:

\[ \mathbb{N} \subset \{0,1,2,\ldots\} \subsetneq \mathbb{R}. \]

- The empty set is a subset of any other set (including itself):

\[ \emptyset \subset \emptyset \subset \{1,2,3\}. \]

- The continuously differentiable real-valued functions on the real line is a subset of the continuous functions:

\[ C^1(\mathbb R, \mathbb R) \subset C(\mathbb R, \mathbb R). \]