Rongzhou at 11:08, 16 October 2026

2026-10-16T11:08:44Z

← Older revision		Revision as of 11:08, 16 October 2026
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	$$		$$

	is the infinite set of all natural number powers of 2 up to $2^20$: 2,4,8,16,32,64,128,256...		is the infinite set of all natural number powers of 2 up to $2^{20}$: $2,4,8,16,32,64,128,256...2^{20}$

	== Special sets ==		== Special sets ==
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	<blockquote>		<blockquote>
	In everyday language, "size" is often used as an alternative term for cardinality.		In everyday language, "size" is often used as an alternative term for cardinality.
			</blockquote>

			If $\|S\|$ is finite, $S$ is known as an '''infinite set'''.

			If $\exists f:\mathbb{Z} \to S$, $S$ is said to be '''countable'''.

			<blockquote>
			All finite sets are countable. Infinite sets may be either countable (e.g., $\mathbb{N}$) or uncountable (e.g., $\mathbb{R}$).
	</blockquote>		</blockquote>

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	If $A \cap B = \emptyset$, $A,B$ are said to be '''disjoint'''.		If $A \cap B = \emptyset$, $A,B$ are said to be '''disjoint'''.

	<gallery class="~~skin-invert-image~~ center">		<gallery class="center">
	File:Venn1010.svg\| $A^c$		File:Venn1010.svg\| $A^c$
	File:Venn0001.svg\| $A \cap B$		File:Venn0001.svg\| $A \cap B$

Rongzhou: Created page with "A '''set''' is an unordered collection of distinct '''elements'''.{{#cite:Q2194}} == Formal definition == According to [https://en.wikipedia.org/wiki/Morse%E2%80%93Kelley_set_theory Morse-Kelley set theory]: $$ \exists B: A \in B \iff A \text{ is a set} $$ {{#cite:Q2184}} == Defining a set == === Enumeration === A small finite set can be described by '''enumerating''' all its elements: $$ S=\{\text{Apple, Banana, Peaches}\} $$ Some infinit..."

2026-10-16T10:51:38Z

Created page with "A '''set''' is an unordered collection of distinct '''elements'''.<ref>{{#cite:Q2194}}</ref> == Formal definition == According to [https://en.wikipedia.org/wiki/Morse%E2%80%93Kelley_set_theory Morse-Kelley set theory]: $$ \exists B: A \in B \iff A \text{ is a set} $$ <ref>{{#cite:Q2184}}</ref> == Defining a set == === Enumeration === A small finite set can be described by '''enumerating''' all its elements: $$ S=\{\text{Apple, Banana, Peaches}\} $$ Some infinit..."

New page

A '''set''' is an unordered collection of distinct '''elements'''.<ref>{{#cite:Q2194}}</ref>

== Formal definition ==

According to [https://en.wikipedia.org/wiki/Morse%E2%80%93Kelley_set_theory Morse-Kelley set theory]:

$$
\exists B: A \in B \iff A \text{ is a set}
$$

<ref>{{#cite:Q2184}}</ref>

== Defining a set ==

=== Enumeration ===

A small finite set can be described by '''enumerating''' all its elements:

$$
S=\{\text{Apple, Banana, Peaches}\}
$$

Some infinite sets following simple patterns can also be enumerated:

$$
S=\{0,1,2,\cdots\} \text{ (Set of natural numbers)}
$$

<blockquote>
To avoid ambiguity, reserve enumeration for simple patterns that most people can identify at first glance. Use [[#Set-builder notation]] elsewise.
</blockquote>

=== Set-builder notation ===

A set that cannot be easily described by enumeration can often be described by rules that all elements of the set follow.

e.g.,

$$
S=\{x | x = 2^n, n \in N\ \land n \le 20\}
$$

is the infinite set of all natural number powers of 2 up to $2^20$: 2,4,8,16,32,64,128,256...

== Special sets ==

{| class="wikitable"
|-
! Set
! Symbol
! Note
|-
| '''Empty set'''
| $\emptyset$
|
|-
| '''Universal set'''
| $U$
| Set of all elements relevant to a particular context
|}

<ref>{{#cite:Q2190}}</ref>

== Set properties ==

=== Cardinality ===

The '''cardinality''' of a set $S$, denoted $|S|$, is its number of elements.

'''Example''':

{| class="wikitable"
|-
! $S$
! | $|S|$
|-
| $\{1,2,3\}$
| 3
|-
| $\{a,b,c,\cdots,z\}$
| 26
|-
| $\{0,2,4,6,8,\cdots\}$
| $\infty$
|}

<blockquote>
In everyday language, "size" is often used as an alternative term for cardinality.
</blockquote>

<ref>{{#cite:Q2196}}</ref>

== Set relationships ==

=== Complement ===

Let $A$ be a set.

$$
x \in A^c \iff x \in U \land x \notin A \text{ (absolute complement)}
$$

=== Intersection, union, symmetric difference ===

Let $A$, $B$ be sets.

$$
x \in A \cap B \iff x \in A \land x \in B \text{ (intersection)} \\
x \in A\cup B \iff x \in A \lor x \in B \text{ (union)} \\
x \in A\triangle B \iff x \in A \oplus x \in B \text{ (symmetric difference)}
$$

If $A \cap B = \emptyset$, $A,B$ are said to be '''disjoint'''.

<gallery class="skin-invert-image center">
File:Venn1010.svg| $A^c$
File:Venn0001.svg| $A \cap B$
File:Venn0111.svg| $A \cup B$
File:Venn0110.svg| $A \triangle B$
</gallery>

<blockquote>
Diagrams like this that visualise sets as circles and represent their relationship via relative positions are called '''Venn diagrams'''.
</blockquote>

<ref>{{#cite:Q2192}}</ref>

=== Subset ===

Let $A$, $B$ be sets.

$$
A \subseteq B \iff \forall x \in A: x \in B \text{ (subset)} \\
A \subset B \iff (A \subseteq B) \land (A \neq B) \text{ (proper subset)}
$$

It can be therefore inferred:

$$
A \not\subseteq B \iff \exists x \in A: x \notin B \\
A \subseteq A \\
\forall S: \emptyset \subseteq S \\
$$

<ref>{{#cite:Q2192}}</ref>

== Direct product ==

See [[direct product (sets)]].

== Power set ==

Let $A$ be a set, $X$ be some sets.

If:

$$
\mathcal{P}(A) = \{X |\forall X \subseteq A\}.
$$

Then

$\mathcal{P}(A)$ is the '''power set''' of $A$.

<blockquote>
Note the power set $\mathcal{P}(A)$ is a set of sets.

'''Example''':

$$
\mathcal{P}(\{a, b\})= \{\emptyset, \{a\}, \{b\}, \{a,b\}\}.
$$
</blockquote>

It can be inferred:

$$
|\mathcal{P}(S)|=2^{|S|}
$$

{{Hidden proof}}
'''Proof by induction''':

Let $S_n:|S_n|=n$ be sets.

By definition of [[#cardinality]], $n \in N$.

If $n=0$:

$$
\mathcal{P}(S_0)=\{\emptyset\} \\
|\mathcal{P}(S_0)|=1=2^n
$$

If $n=1$:

$$
\mathcal{P}(S_1)=\{\emptyset\,A\} \\
|\mathcal{P}(S_1)|=2=2^n
$$

If $n>1$:

Given $|\mathcal{P}(S_n)|=2^n$

For any set $S_{n+1}: |S_{n+1}|=n+1$, identify an arbitrary element $s\in S_{n+1}$, then

$$
S_{n+1}=S_n+s
$$
where set $S_n:|S_n|=n$.

As $|\mathcal{P}(S_n)|=2^n$, $S_n$ has $2^n$ subsets: $A_1=\emptyset,A_2,A_3,...A_{m}$, where $m=2^n$

Since

$$
A_m \subseteq S_n \subset S_{n+1} \Rightarrow A_m \subset S_{n+1}
$$

which gives $m$ subsets of $S_{n+1}$

and

$$
\because s \in S_{n+1} \\
\therefore A_m+s \subseteq S_{n+1}
$$

which gives $m$ more subsets of $S_{n+1}$

Therefore, $S_{n+1}$ has $2m=2 \cdot 2^n=2^{n+1}$ subsets,

i.e.,$|\mathcal{P}(S_{n+1})|=2^{n+1}$

$\therefore $ By induction,

$$
|\mathcal{P}(S)|=2^{|S|}
$$

{{End hidden}}

<ref>{{#cite:Q2192}}</ref>

Set - Revision history

Rongzhou at 11:08, 16 October 2026