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Andrea
2025-10-09 11:43:21 +02:00
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content/set_theory.md
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@@ -37,7 +37,7 @@ Two special sets are defined: the [[empty set|#empty-set]] and the [[universe se
### Empty set ### Empty set
The empty set, denotated 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
@@ -47,7 +47,7 @@ The universal set, denoted by $U$, is a set that has as elements all unique elem
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 denotated 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 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:
@@ -55,13 +55,13 @@ $$ A = \\{ 1, 2, 3, 4, 5, 6, 7 \\} $$
## Element membership ## 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 We denote that an element is a member of a given set with the $\in$ symbol: given the previous set $A$, we write that
$$ 1 \in A $$ $$ 1 \in A $$
This can be read as 1 is a member of set $A$, or 1 is inside set $A$. 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 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
$$ 8 \notin A $$ $$ 8 \notin A $$
@@ -159,7 +159,7 @@ 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 denotated with the $\cup$ symbol. It's denoted with the $\cup$ symbol.
$$ $$
\begin{aligned} \begin{aligned}
@@ -191,7 +191,7 @@ $$
The intersection of two or more sets is a set that contains all elements that belongs in all given sets. The intersection of two or more sets is a set that contains all elements that belongs in all given sets.
It's denotated with the $\cap$ symbol. It's denoted with the $\cap$ symbol.
$$ $$
\begin{aligned} \begin{aligned}
@@ -226,7 +226,7 @@ $$
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 denotated with the $−$ symbol and is formally defined as $A − B = \\{ x ∣ x \in A : x \notin B \\}$ It's denoted with the $−$ symbol and is formally defined as $A − B = \\{ x ∣ x \in A : x \notin B \\}$
$$ $$
\begin{aligned} \begin{aligned}