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content/set_theory.md

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+---
+title: Set Theory
+date: 2024-12-28
+description: Personal notes on Set Theory
+author: nullndr
+tags:
+  - math
+  - notes
+extra:
+  math: true
+banner_image: media/set-theory.webp
+---
+
+# Set Theory
+
+Set theory is the **fundamental block** on which the entire mathematics is built upon. 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: $A \cap A^\complement = \varnothing$.