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Set Theory	2024-12-28	Personal notes on Set Theory	nullndr	math notes	math true	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 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 and the 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 \\} \\\\
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 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.