complement of the set \(A\) (in a given set \(X\))
\(\overline A\)
\(\mathcal P(X)\)
power set of \(X\)
\(\mathcal P(X)\)
\(A\times B\)
cartesian product of the sets \(A\) and \(B\)
\(A\times B\)
\(A\setminus B\)
set difference
\(A\setminus B\)
Let \(X\) be a set. A set \(A\) is called a subset of \(X\) and denoted by \(A\subseteq X\), if each element of \(A\) is also an element of \(X\). Equivalently, X is a superset of \(A\), denoted as \(X\supseteq A\).
If, in addition, \(A\neq X\), then \(A\) is called a proper subset of \(X\) and denoted by \(A\subset X\). Equivalently, X is then a proper superset of \(A\), and denoted as \(X\supset A\).
By convention, we assume that \(\emptyset\subset X\) for any non-empty set \(X\).
Let \(A,B\subseteq X\). The following basic set operators are defined by characterizing the elements of sets. Please note that, according to the extensionality principle, these operations define new sets:
\(A\cup B:=\{x | x\in A\text{ or }x\in B\}\) is the set of all elements \(x\) contained either in \(A\) or in \(B\) and is called the union of the sets \(A\) and \(B\).
\(A\cap B:=\{x | x\in A\text{ and }x\in B\}\) is the set of all elements \(x\) contained both, in \(A\) and in \(B\). It is called the intersection of the sets \(A\) and \(B\).
\(\overline A :=\{x | x\in X\text{ and }x\notin A\}\) is called the complement of the set \(A\) in the set \(X\). It is the set of all elements \(x\) in \(X\) not contained in \(A\).
\(A\setminus B :=\{x | x\in A \text{ and }x\notin B\}\) is called the set difference of the sets \(A\) and \(B\). It is the set of all elements of \(A\) not contained in \(B\).
\(\mathcal P(X) :=\{A | A\subseteq X\}\) is called the power set of the set \(X\). It is the set of all possible subsets of \(X\).
\(A\times B :=\{(x,y)| x\in A \text{ and }x\in B \} \) is called the cartesian product of the sets \(A\) and \(B\). It is the set of (ordered) pairs \((x,y)\), such that the first pair element is contained in \(A\) and the second in \(B\).
Please note that, in general, \(A\times B\neq B\times A\), because the order or the elements changes.
What follows from what?
This is (experimental) work in progress - if you miss an axiom, a definition, a theorem or a proof, if you find any inconsistencies you want to correct, or just know about a cool example or explanation you want to share with others, then join our team and help to improve this catalogue. Learn more about the axiomatic approach on BoP...
