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| title | date | description | author | tags | extra | banner_image | ||||
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| Set Theory | 2024-12-28 | Personal notes on Set Theory | nullndr |
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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.
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 distinct elements, the order of these unique elements does not matter.
These elements are also called members of the set.
The notation of a set is primitive: it can't be defined in terms of 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 etc.
A set can have a finite number of elements, like the books inside a library, or an infinite number of elements.
A set with a finite number of elements is called a finite set, while a set with an infinite number of elements is called an infinite set.
Two special sets are defined: the empty set and the universe set.
Empty set
The empty set, denoted 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.
The universal set depends on the context — it contains all elements under consideration in a given discussion.
Notations
Before going further inside the set theory, it is worth introducing the notations for them.
Sets are typically denoted by an italic capital letter, like A, B, C etc.
To list the elements of a given set the enumeration notation is used. This lists the elements of a set between curly braces. For example, to represent the set A of all natural numbers less than 8 we write the following:
A = \{ 1, 2, 3, 4, 5, 6, 7 \}
We can also use a different notation called set-builder notation, specifying that a set is a set of elements that satisfies some logical formula. More specifically, if P(x) is a logical formula depending on the variable x, then \{x\,|\,P(x)\} denotes the set of all x for which P(x) is true, with the example above we have:
A = \{ x\,|\,x < 8 \}
In this notation, the vertical bar | is read as "such that", and the whole formula can be read as A is the set of all x such that x are less than 8.
Element membership
We denote 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 denotes an element that is not a member of a given set: given the previous set A, we write that
8 \notin A
Indeed the
Aset 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
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)orn(A)
Subsets and super sets
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 super set 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 super set 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 super set 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.
Power set
Given a set A, there exists a set called power set that contains as elements all subsets of A, including the empty set \emptyset and the set A itself. The power set of a set A is denoted as \mathcal{P}(A):
\begin{aligned}
A = \{1, 2, 3\} \\
\mathcal{P}(A) = \{\emptyset, \{1\}, \{2\}, \{3\}, \{1,2\}, \{1,3\}, \{2,3\}, A\}
\end{aligned}
The cardinality of the power set, denoted as |\mathcal{P}(A)| is always 2^{|A|}.
Partitions
A partition of a set A is a set of non-empty subsets of A such that every element a of A belongs to exactly one of these subset (the subsets are non-empty mutually disjoint sets).
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 \text{\,and\,} 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 two 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 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
\vartrianglesymbol 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 \} \\\\
A \times B &= \{ (1,3), (1,4), (2,3), (2,4) \}
\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 matters
The cartesian product of a set by the set itself is denoted by A \times A or A^2.
The absorbing 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.
De Morgan's laws
In propositional logic, De Morgan's laws (also known as De Morgan's theorem), are a pair of transformation rules that are both valid rules of inference. The rules allow the expression of conjunctions and disjunctions purely in term of each other via negation.
\neg (A \land B) = (\neg A) \lor (\neg B)\neg(A \lor B) = (\neg A) \land (\neg B)
In set theory, this gets translated in two ways:
\begin{aligned}
A - (B \cup C) = (A - B) \cap (A - C) \\
A - (B \cap C) = (A - B) \cup (A - C)
\end{aligned}
And
\begin{aligned}
(A \cup B)^\complement = (A^\complement) \cap (B^\complement) \\
(A \cap B)^\complement = (A^\complement) \cup (B^\complement)
\end{aligned}