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12.1: Finite Sets

  • Page ID
    83462
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    Recall.

    For \(m\in\mathbb{N}\) we have defined the counting set

    \begin{equation*} \mathbb{N}_{<m} = \{n \in \mathbb{N} \vert n \lt m\} = \{ 0, \, 1, \, \ldots, \, m-1\} \text{.} \end{equation*}

    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.

    Definition: Finite Set

    a set \(A\) for which there exists a bijection \(\mathbb{N}_{<m} \to A\) for some \(m \in \mathbb{N}\text{,}\) \(m \gt 0\)

    Definition: Cardinality (of a finite set \(A\))

    the unique natural number \(m\) for with a bijection \(\mathbb{N}_{<m} \to A\) exists

    Definition: \(\vert A \vert\)

    the cardinality of the finite set \(A\)

    Definition: \(\text{card} A\)

    alternative notation for the cardinality of the finite set \(A\)

    Definition: \(\#\{\dots\}\)

    alternative notation for the cardinality of the set defined by \(\{\dots\}\)

    Example \(\PageIndex{1}\)

    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.

    Bijections \(\varphi,\psi: \mathbb{N}_{<26} \rightarrow \Sigma\) defined by a table of values.
    \(\sigma\) \(0\) \(1\) \(2\) \(3\) \(\cdots\) \(24\) \(25\)
    \(\varphi(\sigma)\) \(\text{a}\) \(\text{b}\) \(\text{c}\) \(\text{d}\) \(\cdots\) \(\text{y}\) \(\text{z}\)
    \(\psi(\sigma)\) \(\text{a}\) \(\text{z}\) \(\text{b}\) \(\text{y}\) \(\cdots\) \(\text{m}\) \(\text{n}\)

    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?

    Definition: Empty Function

    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.

    Proposition \(\PageIndex{1}\): Properties of empty functions.
    1. For every set \(X\text{,}\) an empty function \(\varnothing \to X\) is injective.
    2. An empty function \(\varnothing \to \varnothing\) is a bijection.
    Proof

    You were asked to verify these statements in Exercise 10.7.12.

    Corollary \(\PageIndex{1}\)

    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.

     

    This page titled 12.1: Finite Sets is shared under a GNU Free Documentation License 1.3 license and was authored, remixed, and/or curated by Jeremy Sylvestre via source content that was edited to the style and standards of the LibreTexts platform.