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@@ -21,38 +21,49 @@ Set theory was born as a necessity to formulate a general theory in order to org
## What is a set? ## What is a set?
A set is a **collection** of unique elements, the order of these unique elements does not matter. 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. 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. > 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 ecc. 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 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__. 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|#empty-set]] and the [[universe set|#universe-set]]. Two special sets are defined: the **empty set** and the **universe set**.
### Empty set ### Empty set
The empty set, denoted with $\\{\\}$ or with the $\varnothing$ symbol, is a special set that has no element. The empty set, denoted with $\{\}$ or with the $\varnothing$ symbol, is a special set that has no element.
### Universe set ### 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, 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 ## Notations
Before going further inside the set theory, it is worth introducing the notations for them. 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. 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 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: 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:
$$ A = \\{ 1, 2, 3, 4, 5, 6, 7 \\} $$ $$ A = \{ 1, 2, 3, 4, 5, 6, 7 \} $$
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 ## 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 We denote that an element is a member of a given set with the $\in$ symbol: given the previous set $A$, we write that
@@ -69,24 +80,24 @@ $$ 8 \notin A $$
This can be read as 8 is not a member of set $A$, or 8 is not inside the set $A$. 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 $$ A = B $$ If two sets $A$ and $B$ contain the same elements we define them as **equal**, denoted by $A = B$:
$$ $$
\begin{aligned} \begin{aligned}
A &= \\{ 1, 2 \\} \\\\ A &= \{ 1, 2 \} \\\\
B &= \\{ 2, 1 \\} \\\\ B &= \{ 2, 1 \} \\\\
A &= B A &= B
\end{aligned} \end{aligned}
$$ $$
---
## Cardinality ## Cardinality
In set theory, the cardinality of a given set $A$ is the number of elements inside the set itself, denoted with $\mid A \mid$. The cardinality of a given set $A$ is the number of elements inside the set itself, denoted with $\mid A \mid$.
$$ $$
\begin{aligned} \begin{aligned}
A &= \\{ \text{blue}, \text{red}, \text{green}, \text{yellow} \\} \\\\ A &= \{ \text{blue}, \text{red}, \text{green}, \text{yellow} \} \\\\
\mid A \mid &= 4 \mid A \mid &= 4
\end{aligned} \end{aligned}
$$ $$
@@ -101,29 +112,31 @@ $$
> The cardinality of a set may also be denoted by $card(A)$ or $n(A)$ > The cardinality of a set may also be denoted by $card(A)$ or $n(A)$
## Subsets and supersets
---
## Subsets and super sets
Between two sets there may exist a relation called **set inclusion**: if all elements inside set $A$ are also elements of $B$, then $A$ is a subset of $B$, denoted with $A \subseteq B$. Between two sets there may exist a relation called **set inclusion**: if all elements inside set $A$ are also elements of $B$, then $A$ is a subset of $B$, denoted with $A \subseteq B$.
$$ $$
\begin{aligned} \begin{aligned}
A &= \\{ 2, 3, 4 \\} \\\\ A &= \{ 2, 3, 4 \} \\\\
B &= \\{ 1, 2, 3, 4, 5 \\} \\\\ B &= \{ 1, 2, 3, 4, 5 \} \\\\
A &\subseteq B A &\subseteq B
\end{aligned} \end{aligned}
$$ $$
When $A$ is a subset of $B$, we can define $B$ as a superset of $A$, denoted with $B \supseteq A$. When $A$ is a subset of $B$, we can define $B$ as a super set of $A$, denoted with $B \supseteq A$.
$$ $$
\begin{aligned} \begin{aligned}
A &= \\{ 2, 3, 4 \\} \\\\ A &= \{ 2, 3, 4 \} \\\\
B &= \\{ 1, 2, 3, 4, 5 \\} \\\\ B &= \{ 1, 2, 3, 4, 5 \} \\\\
B &\supseteq A B &\supseteq A
\end{aligned} \end{aligned}
$$ $$
If the set inclusion relation does not exist between two sets, then we can define $A$ as not being a subset of $B$, formally $A \not\subseteq B$, same for $B$ not being a superset of $A$: $B \not\supseteq A$. If the set inclusion relation does not exist between two sets, then we can define $A$ as not being a subset of $B$, formally $A \not\subseteq B$, same for $B$ not being a super set of $A$: $B \not\supseteq A$.
Given the above definition, a set is always a subset of itself, but there may be cases where we would like to reject this definition. Given the above definition, a set is always a subset of itself, but there may be cases where we would like to reject this definition.
@@ -133,24 +146,45 @@ We denote the fact that a set $A$ is a proper subset of $B$ with the symbol $\su
$$ $$
\begin{aligned} \begin{aligned}
A &= \\{ 2, 3, 4 \\} \\\\ A &= \{ 2, 3, 4 \} \\\\
B &= \\{ 1, 2, 3, 4, 5 \\} \\\\ B &= \{ 1, 2, 3, 4, 5 \} \\\\
A &\subset B A &\subset B
\end{aligned} \end{aligned}
$$ $$
Likewise we denote a set $B$ being a proper superset of $A$ with the symbol $\supset$. Likewise we denote a set $B$ being a proper super set of $A$ with the symbol $\supset$.
$$ $$
\begin{aligned} \begin{aligned}
A &= \\{ 2, 3, 4 \\} \\\\ A &= \{ 2, 3, 4 \} \\\\
B &= \\{ 1, 2, 3, 4, 5 \\} \\\\ B &= \{ 1, 2, 3, 4, 5 \} \\\\
B &\supset A B &\supset A
\end{aligned} \end{aligned}
$$ $$
Note that the empty set $\varnothing$ is a proper subset of any set except itself. Note that the empty set $\varnothing$ is a proper subset of any set except itself.
---
## Power set
Given a set $A$, there exists a set called *power set* that contains as elements all subsets of $A$, including the empty set $\emptyset$ and the set $A$ itself. The power set of a set $A$ is denoted as $\mathcal{P}(A)$:
$$
\begin{aligned}
A = \{1, 2, 3\} \\
\mathcal{P}(A) = \{\emptyset, \{1\}, \{2\}, \{3\}, \{1,2\}, \{1,3\}, \{2,3\}, A\}
\end{aligned}
$$
The cardinality of the power set, denoted as $|\mathcal{P}(A)|$ is always $2^{|A|}$.
---
## Partitions
A *partition* of a set $A$ is a set of non-empty subsets of $A$ such that every element $a$ of $A$ belongs to exactly one of these subset (the subsets are non-empty mutually disjoint sets).
---
# Operations # Operations
Just like algebra has his operations on numbers, sets have their own operations. Just like algebra has his operations on numbers, sets have their own operations.
@@ -159,13 +193,13 @@ Just like algebra has his operations on numbers, sets have their own operations.
The union of two or more sets is a set that contains all elements in the given sets, with no duplicated element. The union of two or more sets is a set that contains all elements in the given sets, with no duplicated element.
It's denoted with the $\cup$ symbol. It's denotated with the $\cup$ symbol.
$$ $$
\begin{aligned} \begin{aligned}
A &= \\{ 2, 4, 6 \\} \\\\ A &= \{ 2, 4, 6 \} \\\\
B &= \\{ 2, 3, 5 \\} \\\\ B &= \{ 2, 3, 5 \} \\\\
A \cup B &= \\{ 2, 3, 4, 5, 6 \\} A \cup B &= \{ 2, 3, 4, 5, 6 \}
\end{aligned} \end{aligned}
$$ $$
@@ -191,13 +225,13 @@ $$
The intersection of two or more sets is a set that contains all elements that belongs in all given sets. The intersection of two or more sets is a set that contains all elements that belongs in all given sets.
It's denoted with the $\cap$ symbol. It's denotated with the $\cap$ symbol.
$$ $$
\begin{aligned} \begin{aligned}
A &= \\{ 2, 4, 6 \\} \\\\ A &= \{ 2, 4, 6 \} \\\\
B &= \\{ 2, 3, 5 \\} \\\\ B &= \{ 2, 3, 5 \} \\\\
A \cap B &= \\{ 2 \\} A \cap B &= \{ 2 \}
\end{aligned} \end{aligned}
$$ $$
@@ -226,13 +260,13 @@ $$
The set difference of two sets $A$ and $B$ is a set that contains all elements of $A$ that are not inside $B$. The set difference of two sets $A$ and $B$ is a set that contains all elements of $A$ that are not inside $B$.
It's denoted with the $−$ symbol and is formally defined as $A − B = \\{ x ∣ x \in A : x \notin B \\}$ It's denotated with the $−$ symbol and is formally defined as $A − B = \{ x ∣ x \in A \text{\,and\,} x \notin B \}$
$$ $$
\begin{aligned} \begin{aligned}
A &= \\{ 2, 4, 6 \\} \\\\ A &= \{ 2, 4, 6 \} \\\\
B &= \\{ 2, 3, 5 \\} \\\\ B &= \{ 2, 3, 5 \} \\\\
A - B &= \\{ 4, 6 \\} A - B &= \{ 4, 6 \}
\end{aligned} \end{aligned}
$$ $$
@@ -240,12 +274,12 @@ The set difference is not commutative, indeed
$$ $$
\begin{aligned} \begin{aligned}
A - B &= \\{ 4, 6 \\} \\\\ A - B &= \{ 4, 6 \} \\\\
B - A &= \\{ 3, 5 \\} B - A &= \{ 3, 5 \}
\end{aligned} \end{aligned}
$$ $$
The set difference has three absorbing elements: The set difference has two absorbing elements:
$$ $$
\begin{aligned} \begin{aligned}
@@ -263,15 +297,15 @@ $$
## Symmetric Difference ## 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. 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 $B − A$ It's defined as the union of the set difference $A − B$ with the set difference $B − A$
$$ $$
\begin{aligned} \begin{aligned}
A &= \\{ 1, 2, 5, 6 \\} \\\\ A &= \{ 1, 2, 5, 6 \} \\\\
B &= \\{ 2, 3, 4, 7 \\} \\\\ B &= \{ 2, 3, 4, 7 \} \\\\
(A − B) \cup (B − A) &= \\{ 1, 3, 4, 5, 6, 7 \\} (A − B) \cup (B − A) &= \{ 1, 3, 4, 5, 6, 7 \}
\end{aligned} \end{aligned}
$$ $$
@@ -281,8 +315,8 @@ The symmetric difference is both commutative and associative
$$ $$
\begin{aligned} \begin{aligned}
A &= \\{ 1, 2, 5, 6 \\} \\\\ A &= \{ 1, 2, 5, 6 \} \\\\
B &= \\{ 2, 3, 4, 7 \\} \\\\ B &= \{ 2, 3, 4, 7 \} \\\\
A \vartriangle B &= B \vartriangle A \\\\ A \vartriangle B &= B \vartriangle A \\\\
(A \vartriangle B) \vartriangle C &= A \vartriangle (B \vartriangle C) (A \vartriangle B) \vartriangle C &= A \vartriangle (B \vartriangle C)
\end{aligned} \end{aligned}
@@ -304,13 +338,13 @@ $$
The cartesian product of two sets $A$ and $B$, denoted by $A \times B$, is a set of ordered pairs $(a,b)$ where $a$ is an element of $A$, and $b$ is an element of $B$. The cartesian product of two sets $A$ and $B$, denoted by $A \times B$, is a set of ordered pairs $(a,b)$ where $a$ is an element of $A$, and $b$ is an element of $B$.
Formally it is defined as $A \times B = \\{ (a,b) ∣ a \in A, b \in B \\}$. Formally it is defined as $A \times B = \{ (a,b) ∣ a \in A, b \in B \}$.
$$ $$
\begin{aligned} \begin{aligned}
A &= \\{ 1, 2 \\} \\\\ A &= \{ 1, 2 \} \\\\
B &= \\{ 3, 4 \\} \\\\ B &= \{ 3, 4 \} \\\\
A \times B &= \\{ (1,3), (1,4), (2,3), (2,4) \\} A \times B &= \{ (1,3), (1,4), (2,3), (2,4) \}
\end{aligned} \end{aligned}
$$ $$
@@ -318,18 +352,18 @@ The cartesian product is not commutative, because $A \times B$ is not the same a
$$ $$
\begin{aligned} \begin{aligned}
A &= \\{ 1, 2 \\} \\\\ A &= \{ 1, 2 \} \\\\
B &= \\{ 3, 4 \\} \\\\ B &= \{ 3, 4 \} \\\\
A \times B &= \\{ (1,3), (1,4), (2,3), (2,4) \\} \\\\ A \times B &= \{ (1,3), (1,4), (2,3), (2,4) \} \\\\
B \times A &= \\{ (3,1), (3,2), (4,1), (4,2) \\} B \times A &= \{ (3,1), (3,2), (4,1), (4,2) \}
\end{aligned} \end{aligned}
$$ $$
> Remember that the order inside a pair matter > Remember that the order inside a pair matters
The cartesian produt of a set by the set itself is denoted by $A \times A$ or $A^2$. The cartesian product of a set by the set itself is denoted by $A \times A$ or $A^2$.
The absobing element of the cartesian product is the $\varnothing$ set The absorbing element of the cartesian product is the $\varnothing$ set
$$ $$
A \times \varnothing = \varnothing A \times \varnothing = \varnothing
@@ -339,13 +373,13 @@ $$
The complement of a set $A$, denoted by $A^\complement$ is the set difference of the universal set $U$ with $A$. 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 $A^\complement = \\{ x ∣ x \in U \land x \not\in A \\}$ It is formally defined as $A^\complement = \{ x ∣ x \in U \land x \not\in A \}$
$$ $$
\begin{aligned} \begin{aligned}
U &= \\{ 1, 2, 3, 4, 5, 6, 7, 8, 9 \\} \\\\ U &= \{ 1, 2, 3, 4, 5, 6, 7, 8, 9 \} \\\\
A &= \\{ 1, 3, 5, 7, 9 \\} \\\\ A &= \{ 1, 3, 5, 7, 9 \} \\\\
A^\complement = U − A &= \\{ 2, 4, 6, 8 \\} A^\complement = U − A &= \{ 2, 4, 6, 8 \}
\end{aligned} \end{aligned}
$$ $$
@@ -353,4 +387,30 @@ The complement of the complement a set $A$, denoted by $(A^\complement)^\complem
The union of set $A$ with the complement of itself results in the universal set $U$: $A \cup A^\complement=U$. 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: $A \cap A^\complement = \varnothing$. 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.
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
$$
\begin{aligned}
(A \cup B)^\complement = (A^\complement) \cap (B^\complement) \\
(A \cap B)^\complement = (A^\complement) \cup (B^\complement)
\end{aligned}
$$