# 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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Concept | Content |
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Logical Statements and Operations | Logic is the foundation to formulate proofs and to understand the language of mathematics. |

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Concept | Content |
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Maps | Maps are the mathematical formulation of a machine that gets inputs and generate outputs. On both sides, sets are needed for the domain and the codomain. |

Real Numbers | In a real analysis, the real numbers are the largest number set we need. They satisfy axioms that represent the idea of a number line. |

Natural Numbers and Induction | Using natural numbers is our first mathematical abstraction as children. Mathematical induction is an important technique of proof. |

Image and Preimage | Via images and preimages we describe how functions work on sets. |

Bounded Sets, Maxima and Minima | The values inside a set of real numbers can be bounded. |

Supremum and Infimum of Sets | Bounded sets always have an supremum and infimum which are generalizations of maximum and minimum. |

Open, Closed, Compact Sets | Important notions for subsets of real numbers. |

Interior, Closure, Boundary | Topological operations on sets. |

Sequences of Bounded Functions | The concept of sequences but for functions instead of real numbers. |

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

## Study *Sets* #

Modern mathematics does not say what sets are, but only specifies rules. This is, however, too difficult for us right now, and we rather cite the attempt of a definition by Georg Cantor:

**Concept 1**. “Unter einer ‚Menge‘ verstehen wir jede
Zusammenfassung von bestimmten wohlunterschiedenen Objekten unserer
Anschauung oder unseres Denkens zu einem Ganzen.”

**Definition 2** (Set, element). *A
set is a collection into a whole of definite, distinct objects
of our perception or of our thought. Such an object \(x\) of a set \(M\) is called an element of \(M\) and one writes \(x\in M\). If \(x\) is not such an object of \(M\), we write \(x\not\in M\).*

A set is defined by giving all its elements \(M:=\{1,4,9\}\).

**Concept 3**. The symbol “\(:=\)” is read as and means that the symbol
\(M\) is newly introduced as a set by
the given elements.

**Example 4**.

The empty set \(\{\} = \emptyset = \varnothing\) is the unique set that has no elements at all.

The set that contains the empty set \(\{ \varnothing \}\), which is non-empty since it has exactly one element.

A finite set of numbers is \(\{ 1,2,3\}\).

**Notation 5**. Let \(A,B\) be sets:

\(x \in A\) means \(x\) is an element of \(A\)

\(x\not\in A\) means \(x\) is not an element of \(A\)

\(A \subset B\) means \(A\) is a subset of \(B\): every element of \(A\) is contained in \(B\)

\(A \supset B\) means \(A\) is a superset of \(B\): every element of \(B\) is contained in \(A\)

\(A=B\) means \(A \subset B \wedge A \supset B\). Note that the order of the elements does not matter in sets. If we want the order to matter, we rather define

*tuples*: \((1,2,3) \neq (1,3,2)\). For sets, we always have \(\{ 1,2,3 \} = \{1, 3,2\}\).\(A \subsetneq B\) means \(A\) is a “proper” subset of \(B\), every element of \(A\) is contained in \(B\), but \(A \neq B\).

**Concept 6** (The important number sets).

\(\mathbb{N}\) is the set of the natural numbers \(1,2,3,\ldots\);

\(\mathbb{N}_0\) is the set of the natural numbers and zero: \(0, 1, 2, 3,\dots\);

\(\mathbb{Z}\) is the set of the integers, which means \(\ldots,-3,-2,-1,0,1,2,3,\ldots\);

\(\mathbb{Q}\) is the set of the rational numbers, which means all fractions \(\frac pq\) with \(p\in\mathbb{Z}\) and \(q\in\mathbb{N}\);

\(\mathbb{R}\) is the set of the real numbers.

Other ways to define sets: \[\begin{aligned} A &= \{ n \in \mathbb{N} : 1 \le n \le 300 \}\\ \mathbf{P}(B) &= \{ M : M \subset B \} \mbox{ power set: set of all subsets of } B\\ I &= \{ x\in \mathbb{R} : 1 \le x < \pi \} = [1,\pi) \mbox{ half-open interval } \end{aligned}\]

**Definition 7** (Cardinality). *We use vertical bars
\(|\cdot|\) around a set to denote the
number of elements. For example, we have \(|\{1,4,9\}|=3\). The number of elements is
called the cardinality of the set.*

**Example 8**. \(|\{1,3,3,1\}|=2\), \(\quad |\{1,2,3, \ldots, n \}| = n\), \(\quad | \mathbb{N} | = \infty\) (?)

Courtesy of Marcus Waurick. *Well-defined & Wonderful podcast*, marcus-waurick.de.

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