12.1: Finite Sets
- Page ID
- 83462
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Recall.
For \(m\in\mathbb{N}\) we have defined the counting set
Clearly, \(\mathbb{N}_{<m}\) contains exactly \(m\) elements. In fact, we have defined the number \(m\) to be the set \(\mathbb{N}_{<m}\text{.}\) (See Example 11.4.2.)
As the terminology implies, we will use these sets to count the elements of other sets. In particular, given a set \(A\text{,}\) if we can match up the elements of \(A\) with the elements of \(\mathbb{N}_{<m}\text{,}\) one for one, then \(A\) must also contain exactly \(m\) elements.
a set \(A\) for which there exists a bijection \(\mathbb{N}_{<m} \to A\) for some \(m \in \mathbb{N}\text{,}\) \(m \gt 0\)
For finite set \(A\) there exists one unique natural number \(m\) for which a bijection \(\mathbb{N}_{<m} \to A\) exists.
Suppose \(A\) is finite. While there is only one number \(m\) for which a bijection \(\mathbb{N}_{<m} \to A\) exists, there can be many such bijections, and the number of bijections increases as \(m\) increases.
Prove Fact \(\PageIndex{1}\).
the unique natural number \(m\) for with a bijection \(\mathbb{N}_{<m} \to A\) exists
the cardinality of the finite set \(A\)
alternative notation for the cardinality of the finite set \(A\)
alternative notation for the cardinality of the set defined by \(\{\dots\}\)
For \(\Sigma = \{a, b, \ldots, z\} \text{,}\) we have \(\vert \Sigma \vert = 26\text{.}\) Below are two example bijections \(\varphi,\psi: \mathbb{N}_{<26} \rightarrow \Sigma\) that verify this cardinality number.
\(\sigma\) | \(0\) | \(1\) | \(2\) | \(3\) | \(\cdots\) | \(24\) | \(25\) |
\(\varphi(\sigma)\) | \(\text{a}\) | \(\text{b}\) | \(\text{c}\) | \(\text{d}\) | \(\cdots\) | \(\text{y}\) | \(\text{z}\) |
\(\psi(\sigma)\) | \(\text{a}\) | \(\text{z}\) | \(\text{b}\) | \(\text{y}\) | \(\cdots\) | \(\text{m}\) | \(\text{n}\) |
Cardinality of an empty set.
What about the empty set? Clearly we should have \(\vert \varnothing \vert = 0\text{.}\) But is this consistent with our definition of cardinality?
a function with domain \(\varnothing\)
If we accept the existence of empty functions \(\varnothing \to X\) for every set \(X\text{,}\) then the properties of such functions that we need in order to establish \(\vert \varnothing \vert = 0\) will be vacuously true.
- For every set \(X\text{,}\) an empty function \(\varnothing \to X\) is injective.
- An empty function \(\varnothing \to \varnothing\) is a bijection.
- Proof
-
You were asked to verify these statements in Exercise 10.7.12.
The cardinality of the empty set is \(0\text{.}\)
- Proof.
-
We are required to demonstrate an example of a bijection \(\mathbb{N}_{<0} \to \varnothing\text{.}\) But
\begin{equation*} \mathbb{N}_{<0} = \{n \in \mathbb{N} \vert n \lt 0 \} = \varnothing \text{,} \end{equation*}
so Statement 2 of Proposition \(\PageIndex{1}\) tells that the empty function \(\mathbb{N}_{<0} \to \varnothing\) is indeed a bijection.