Sets can be joined or intersected in order to create new sets.

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Concept |
Content |

Sets |
Sets are the basic building blocks for a lot of mathematics. In order to rigorously define numbers and doing real analysis, we need to know how to work with sets. |

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We remember the important operations for sets:

\(M_1 \cup M_2 := \{ x \mid x\in M_1
\vee x \in M_2 \}\) (union)

\(M_1 \cap M_2 := \{ x \mid x \in M_1
\wedge x \in M_2\}\) (intersection)

\(M_1 \setminus M_2 := \{ x \mid x \in
M_1 \wedge x \not\in M_2 \}\) (set difference)

**Definition 1** (Set compositions). *The
**union* \(M_1\cup M_2\) is the
new set that consists exactly of the objects that are elements of \(M_1\) **or** \(M_2\).

*The **intersection* \(M_1\cap
M_2\) is the new set whose elements are the objects that are
elements of \(M_1\)
**and** \(M_2\).

*We write \(M_1\setminus M_2\)
for the **set difference* whose elements are the objects that are
elements of \(M_1\) **but
not** elements of \(M_2\).

*A **subset* of \(M_2\) is
each set whose elements are also elements of \(M_2\).

**Definition 2** (Complement set). *Let \(X\) be a set. Then for a subset \(M \subset X\) there is a unique
**complement* of \(M\) with
respect to \(X\): \[M^c := X \setminus M = \{ x \in X \mid x \notin M
\}\]

**Definition 3** (Product set, Cartesian product).
*The **Cartesian product* of two sets \(A,B\) is given as the set of all
*pairs* (two elements with order): \[A
\times B
:= \{ (a,b) \mid a \in A, b \in B\}\] In the same sense,
for sets \(A_1, \ldots, A_n\) the set
of all *\(n\)-tupels* is
defined: \[A_1 \times \cdots \times
A_n
:= \{ (a_1,\ldots, a_n) \mid a_1 \in A_1, \ldots, a_n \in
A_n\}\]

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