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