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.

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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.

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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 $A$ 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 $A = B$:

$$ \begin{aligned} A &= { 1, 2 } \\ B &= { 2, 1 } \\ A &= B \end{aligned} $$

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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 $card(A)$ or $n(A)$

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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.

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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 $2^{|A|}$.

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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).

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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 $\vartriangle$ 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 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: $A \cap A^\complement = \varnothing$.

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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.

1. $\neg (A \land B) = (\neg A) \lor (\neg B)$
2. $\neg(A \lor B) = (\neg A) \land (\neg B)$

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

$$ \begin{aligned} (A \cup B)^\complement = (A^\complement) \cap (B^\complement) \ (A \cap B)^\complement = (A^\complement) \cup (B^\complement) \end{aligned} $$