1.1 Sets and Functions

2.1 The Algebraic and Order Properties of R >>

For brevity, we implicitly let for all sections in this page

• $x$ : Element
• $A,B,C$ : Set

Basic set notations

Notation	Meaning
$x\in A$	$x$ is in $A$; $x$ is a member of $A$; $x$ belongs to $A$
$x\notin A$	$x$ not in $A$;
$A\subseteq B$; $B\supseteq A$	Every element of $A$ also belongs to $B$; $A$ is a subset of $B$
$A\subset B$; $B\supset A$	$A$ contains some element of $B$, but not all of $B$; $A$ is a proper subset of $B$

1.1.1 Definition: Set equality

Definition

Let $A,B$ : Set

If $A,B$ contains the same elements

Then

• We say $A$ and $B$ are equal
• We write $A=B$

Thus, to prove that the sets $A$ and $B$ are equal, we must show that $A\subseteq B$ and $B\subseteq A$.

1.1.3 Definition: Set union, intersection, complement

Definition

We say the set $A\cup B:=\{x:x\in A\text{ or }x\in B\}$ is the union of $A$ and $B$

Let $n\in \mathbb{N}$. $\{A_1,A_2,\dots ,A_n,\dots \}$ : Infinite collection of sets. Their union is the set of elements that belong to at least one of the sets $A_n$ :

$$\bigcup _{n=1}^{\infty }{A}_n:=\{x:x\subset A_n\}$$

Definition

We say the set $A\cap B:=\{x:x\in A\text{ and }x\in B\}$ is the intersection of $A$ and $B$

Let $n\in \mathbb{N}$. $\{A_1,A_2,\dots ,A_n,\dots \}$ : Infinite collection of sets. Their union is the set of elements that belong to all of these sets $A_n$ : $ \bigcap_{n=1}^\infty A_{n} := { x:x \subset A_{n} } $$

Definition

If $ A - B:={ x:x \in A\text{ and } x \not \in B } $$

Then we say set $A-B$ is the complement of $B$ relative to $A$

Remark

It is also often denoted by $A\setminus B$.

Or simply:

• Union: All elements existing in $A$, but not existing in $B$
• Intersection: All elements existing in either $A$ or $B$
• Complement: All elements existing in both $A$ and $B$ at the same time

Definition: Empty set

Definition: Disjoint sets

1.1.4 Theorem

$$\begin{array}{rl}A\setminus (B\cup C)& =(A\setminus B)\cap (A\setminus C)\\ A\setminus (B\cap C)& =(A\setminus B)\cup (A\setminus C)\end{array}$$

See also

• 1.2 Sets