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It has been generalized only in 1870s from [Georg Cantor](https://en.wikipedia.org/wiki/Georg_Cantor). + +Even if the set theory is a quite recently mathematical branch, it preceeds all other branches: arithmetic studies the set of natural number; algebra studies the set of rational and real numbers and geometry studies sets of points, like segments. + +Set theory was born as a necessity to formulate a general theory in order to organize all mathematicals theories. + +## What is a set? + +A set is a **collection** of unique elements, the order of these unique elements does not matter. + +These elements are also called **members** of the set. + +> The set concept is __primitive__: it is not possible to define it with more simpler concepts. + +We can find sets in our every day life, for example the set of books a library has, the set of libraries in a city, the set of cities in a country ecc. + +A set can have a finite number of elements, like the books inside a library, or an infinite number of elements. + +A set with an infinite number of elements is called an __infinite set__. + +Two special sets are defined: the [[empty set|#empty-set]] and the [[universe set|#universe-set]]. + +### Empty set + +The empty set, denotated with $\\{\\}$ or with the $\varnothing$ symbol, is a special set that has no element. + +### Universe set + +The universal set, denoted by $U$, is a set that has as elements all unique elements of all related sets. + +## Notations + +Before going further inside the set theory, it is worth introducing the notations for them. + +Sets are typically denotated by an italic capital letter, like $A$, $B$, $C$ etc. + +To list the elements of a given set the enumeration notation is used. This list the elements of a set between curly braces. For example, to reppresent the set $A$ of all natural numbers less than 8 we write the following: + +$$ A = \\{ 1, 2, 3, 4, 5, 6, 7 \\} $$ + +## Element membership + +We denotate that an element is a member of a given set with the $\in$ symbol: given the previous set $A$, we write that + +$$ 1 \in A $$ + +This can be read as 1 is a member of set $A$, or 1 is inside set $A$. + +The symbol $\notin$ is the opposite of the $\in$ symbol, it denotates an element that is not a member of a given set: given the previous set $A$, we write that + +$$ 8 \notin A $$ + +> Indeed the $A$ set definition was "the set of all natural numbers less than 8" + +This can be read as 8 is not a member of set $A$, or 8 is not inside the set $A$. + +If two sets $A$ and $B$ contain the same elements we define them as **equal**, denoted by $$ A = B $$ + +$$ +\begin{aligned} +A &= \\{ 1, 2 \\} \\\\ +B &= \\{ 2, 1 \\} \\\\ +A &= B +\end{aligned} +$$ + + +## Cardinality + +In set theory, the cardinality of a given set $A$ is the number of elements inside the set itself, denoted with $\mid A \mid$. + +$$ +\begin{aligned} +A &= \\{ \text{blue}, \text{red}, \text{green}, \text{yellow} \\} \\\\ +\mid A \mid &= 4 +\end{aligned} +$$ + +The empty set $\varnothing$ has a cardinality of $0$. + +$$ +\begin{aligned} +\mid \varnothing \mid &= 0 +\end{aligned} +$$ + +> The cardinality of a set may also be denoted by $card(A)$ or $n(A)$ + +## Subsets and supersets + +Between two sets there may exist a relation called **set inclusion**: if all elements inside set $A$ are also elements of $B$, then $A$ is a subset of $B$, denoted with $A \subseteq B$. + +$$ +\begin{aligned} +A &= \\{ 2, 3, 4 \\} \\\\ +B &= \\{ 1, 2, 3, 4, 5 \\} \\\\ +A &\subseteq B +\end{aligned} +$$ + +When $A$ is a subset of $B$, we can define $B$ as a superset of $A$, denoted with $B \supseteq A$. + +$$ +\begin{aligned} +A &= \\{ 2, 3, 4 \\} \\\\ +B &= \\{ 1, 2, 3, 4, 5 \\} \\\\ +B &\supseteq A +\end{aligned} +$$ + +If the set inclusion relation does not exist between two sets, then we can define $A$ as not being a subset of $B$, formally $A \not\subseteq B$, same for $B$ not being a superset of $A$: $B \not\supseteq A$. + +Given the above definition, a set is always a subset of itself, but there may be cases where we would like to reject this definition. + +For these cases we can define the term __proper subset__: the set $A$ is a proper subset of $B$ only if $A$ is a subset of $B$ and $A$ is not equal to $B$. + +We denote the fact that a set $A$ is a proper subset of $B$ with the symbol $\subset$. + +$$ +\begin{aligned} +A &= \\{ 2, 3, 4 \\} \\\\ +B &= \\{ 1, 2, 3, 4, 5 \\} \\\\ +A &\subset B +\end{aligned} +$$ + +Likewise we denote a set $B$ being a proper superset of $A$ with the symbol $\supset$. + +$$ +\begin{aligned} +A &= \\{ 2, 3, 4 \\} \\\\ +B &= \\{ 1, 2, 3, 4, 5 \\} \\\\ +B &\supset A +\end{aligned} +$$ + +Note that the empty set $\varnothing$ is a proper subset of any set except itself. + +# Operations + +Just like algebra has his operations on numbers, sets have their own operations. + +## Union + +The union of two or more sets is a set that contains all elements in the given sets, with no duplicated element. + +It's denotated with the $\cup$ symbol. + +$$ +\begin{aligned} +A &= \\{ 2, 4, 6 \\} \\\\ +B &= \\{ 2, 3, 5 \\} \\\\ +A \cup B &= \\{ 2, 3, 4, 5, 6 \\} +\end{aligned} +$$ + +The union is both commutative and associative + +$$ +\begin{aligned} +A \cup B &= B \cup A \\\\ +A \cup (B \cup C) &= (A \cup B) \cup C +\end{aligned} +$$ + +and has two neutral elements: the $\varnothing$ set and the set itself + +$$ +\begin{aligned} +A \cup \varnothing &= A \\\\ +A \cup A &= A +\end{aligned} +$$ + +## Intersection + +The intersection of two or more sets is a set that contains all elements that belongs in all given sets. + +It's denotated with the $\cap$ symbol. + +$$ +\begin{aligned} +A &= \\{ 2, 4, 6 \\} \\\\ +B &= \\{ 2, 3, 5 \\} \\\\ +A \cap B &= \\{ 2 \\} +\end{aligned} +$$ + +Like union, the intersection is both commutative and associative + +$$ +\begin{aligned} +A \cap B &= B \cap A \\\\ +A \cap (B \cap C) &= (A \cap B) \cap C +\end{aligned} +$$ + +It has only one neutral element: the set itself + +$$ +A \cap A = A +$$ + +While the $\varnothing$ set is the absorbing element of the intersection, because the intersection of any set with the $\varnothing$ set results in the $\varnothing$ set + +$$ +A \cap \varnothing = \varnothing +$$ + +## Set Difference + +The set difference of two sets $A$ and $B$ is a set that contains all elements of $A$ that are not inside $B$. + +It's denotated with the $−$ symbol and is formally defined as $A − B = \\{ x ∣ x \in A : x \notin B \\}$ + +$$ +\begin{aligned} +A &= \\{ 2, 4, 6 \\} \\\\ +B &= \\{ 2, 3, 5 \\} \\\\ +A - B &= \\{ 4, 6 \\} +\end{aligned} +$$ + +The set difference is not commutative, indeed + +$$ +\begin{aligned} +A - B &= \\{ 4, 6 \\} \\\\ +B - A &= \\{ 3, 5 \\} +\end{aligned} +$$ + +The set difference has three absorbing elements: + +$$ +\begin{aligned} +A - A &= \varnothing \\\\ +A - U &= \varnothing +\end{aligned} +$$ + +The $\varnothing$ set is both the neutral element and one of the absorbing element of the set difference + +$$ +A - \varnothing = A \\\\ +\varnothing - A = \varnothing +$$ + +## Symmetric Difference + +The symmetric difference of two sets or more sets, also called __disjunctive union__, is a set that contains all unique elements of a set that are not inside the other sets. + +It's defined as the union of the set difference $A − B$ with the set difference $B − A$ + +$$ +\begin{aligned} +A &= \\{ 1, 2, 5, 6 \\} \\\\ +B &= \\{ 2, 3, 4, 7 \\} \\\\ +(A − B) \cup (B − A) &= \\{ 1, 3, 4, 5, 6, 7 \\} +\end{aligned} +$$ + +> The $\vartriangle$ symbol is also used to represent symmetric difference + +The symmetric difference is both commutative and associative + +$$ +\begin{aligned} +A &= \\{ 1, 2, 5, 6 \\} \\\\ +B &= \\{ 2, 3, 4, 7 \\} \\\\ +A \vartriangle B &= B \vartriangle A \\\\ +(A \vartriangle B) \vartriangle C &= A \vartriangle (B \vartriangle C) +\end{aligned} +$$ + +The $\varnothing$ set is the neutral element of the symmetric difference + +$$ +A \vartriangle \varnothing = A +$$ + +While the absorbing element is the set itself + +$$ +A \vartriangle A = \varnothing +$$ + +## Cartesian Product + +The cartesian product of two sets $A$ and $B$, denoted by $A \times B$, is a set of ordered pairs $(a,b)$ where $a$ is an element of $A$, and $b$ is an element of $B$. + +Formally it is defined as $A \times B = \\{ (a,b) ∣ a \in A, b \in B \\}$. + +$$ +\begin{aligned} +A &= \\{ 1, 2 \\} \\\\ +B &= \\{ 3, 4, 5 \\} \\\\ +A \times B &= \\{ (1,3), (1,4), (1,5), (2,3), (2,4), (2,5) \\} +\end{aligned} +$$ + +The cartesian product is not commutative, because $A \times B$ is not the same as $B \times A$ + +$$ +\begin{aligned} +A &= \\{ 1, 2 \\} \\\\ +B &= \\{ 3, 4 \\} \\\\ +A \times B &= \\{ (1,3), (1,4), (2,3), (2,4) \\} \\\\ +B \times A &= \\{ (3,1), (3,2), (4,1), (4,2) \\} +\end{aligned} +$$ + +> Remember that the order inside a pair matter + +The cartesian produt of a set by the set itself is denoted by $A \times A$ or $A^2$. + +The absobing element of the cartesian product is the $\varnothing$ set + +$$ +A \times \varnothing = \varnothing +$$ + +## Complement + +The complement of a set $A$, denoted by $A^\complement$ is the set difference of the universal set $U$ with $A$. + +It is formally defined as $A^\complement = \\{ x ∣ x \in U \land x \not\in A \\}$ + +$$ +\begin{aligned} +U &= \\{ 1, 2, 3, 4, 5, 6, 7, 8, 9 \\} \\\\ +A &= \\{ 1, 3, 5, 7, 9 \\} \\\\ +A^\complement = U − A &= \\{ 2, 4, 6, 8 \\} +\end{aligned} +$$ + +The complement of the complement a set $A$, denoted by $(A^\complement)^\complement$ is the set $A$ itself. + +The union of set $A$ with the complement of itself results in the universal set $U$: $A \cup A^\complement=U$. + +The intersection of set $A$ with the complement of itself results in the $\varnothing$ set: $A \cap A^\complement = \varnothing$. \ No newline at end of file