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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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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