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