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## Some elements of set theory

We start with elementary facts about sets, but we do not try to put set theory on an axiomatic basis. Axiomatic set theory puts the construction of sets on an axiomatic base, to avoid paradoxes which result from defining large, inclusive sets. One famous example which caused a shockwave through the mathematical society was the discovery of Bertrand Russell in 1901 that the set theory created by George Cantor leads to a contradiction: $\text{Let } R=\{x | x \notin x \} \text{ Then } R \in R \Leftrightarrow R \notin R$ Modifications to set theory proposed in the 1920s by Abraham Fraenkel, Thoralf Skolem, and by Ernst Zermelo resulted in the axiomatic set theory called ZFC. This theory became widely accepted once Zermelo's axiom of choice (the C in ZFC) ceased to be controversial. ZFC has remained the canonical axiomatic set theory down to the present day.
For the developing of calculus this more formal form of set theory is not needed. Therefore we do not give a precise definition of a set, a set is an undefinable object like points and lines in the theory of Euclidean geometry. The objects we deal with are either sets or primitive. This is in contrast with ZFC where only sets exists. We assume all primitive objects belong to a universe $U$. Objects have properties and relations with other objects. We use symbols (mainly letters) to denote objects, properties and relations. For sets we use upper case, for primitives we use lower case. Realize that at this stage of our discussion we have not yet defined the concept of a number and counting.

### Sets

$x=y$
the object denoted by the symbols $x$ and $y$ are the same
$x \neq y$
the object denoted by the symbols $x$ and $y$ are not the same
$x \in X$
$x$ is an element of the set $X$
$x \notin X$
$x$ is not an element of the set $X$
$X \subset Y$
$X$ is a subset of $Y$ which means $\forall x (x \in X \Rightarrow Y)$
$X \not \subset Y$
$X$ is a not subset of $Y$ which means $\lnot (\forall x (x \in X \Rightarrow Y))$
$X = Y$
two sets $X$ and $Y$ are equal if and only if they have the same elements, $X \subset Y \land Y \subset X$.
$\{x \in X | P(x)\}$
a unique subset of $X$ such that for all elements $x \in X$ $P(x)=true$
$\{x \in X | P(x)\} \subset \{x \in X | Q(x) \}$
$(\forall x \in X)\left(P(x) \Rightarrow Q(x)\right)$
$\{x \in X | P(x)\} = \{x \in X | Q(x) \}$
$(\forall x \in X)\left(P(x) \Leftrightarrow Q(x)\right)$
$\emptyset_X$
the empty set of $X$, $\{x \in X | x\neq x\}$
$\emptyset$
Let $X$ and $Y$ be any two sets then $(x \in \emptyset_X \Rightarrow P(x))$ for any $P(x)$, because a false statement implies every statement (this follows from: $p \Rightarrow q \Leftrightarrow \lnot p \lor q)$. We have now $(x \in \emptyset_X) \Rightarrow (x \in \emptyset_Y)$ and also $(x\in \emptyset_Y) \Rightarrow (x \in \emptyset_X)$ from which follows $\emptyset_X = \emptyset_Y$. All empty sets are equal and denoted by $\emptyset$.
$\{a_1,a_2,...,a_n \}$
the set $A$ consists of elements $x \in A$ if and only if $x=a_i$ for some $i=1,2,...,n$.
$\{ a \}$
the set having object $a$ as unique element
$\mathcal{A}$
an non-indexed family of sets, whose elements are sets
$\{A_\alpha\}_{\alpha \in \Lambda}$
an indexed family of sets, whose elements are sets indexed by an indexing set $\Lambda$, $\Lambda \neq \emptyset$
$\mathcal{P}(X)$
the powerset of $X$ consists of all subsets of $X$, $x \in X \Leftrightarrow \{x\} \in \mathcal{P}(x)$

### Set operations

$X \setminus Y$ or $\complement_X Y$