# 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:

$C^1(\mathbb R, \mathbb R) \subset C(\mathbb R, \mathbb R).$