Size of a set

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Subsection 2.1.3 Size of a set

We give an intuitive definition of the number of elements in a set.

Definition 2.1.3.1.

Let \(A\) be a set. If there are exactly \(n\) distinct elements in \(A\text{,}\) where \(n\) is a non-negative integer, we say \(A\) is a finite set and that \(n\) is the cardinality of \(A\text{,}\) denoted \(\size{A}\text{.}\)

We give a precise definition of cardinality in Definition 4.1.1.2. In Section 4.1 we also discuss sets that are not finite and their cardinality The notion if infinity is subtle; there are different sizes of infinity.

Example 2.1.3.2. The cardinality of the empty set is zero.

The empty set \(\emptyset\) has size \(0\)

\begin{equation*} \size{\emptyset} = 0 \end{equation*}

since \(\set{}\) has no elements.

Example 2.1.3.3. Cardinality of a set in roster form.

Let \(E\) be the even integers between \(-5\) and \(5\text{,}\) not including \(-5\) and \(5\text{.}\) Then we can compute the size of \(E\) by writing \(E\) using the roster method and counting.

\begin{equation*} E = \set{-4, -2, 0, 2, 4}, \end{equation*}

so \(\size{E} = 5\text{.}\)

Example 2.1.3.4. Number of consonants.

Let \(C\) be the set of consonants in the English alphabet. Instead of listing the elements of \(C\) in roster method, we note that there are \(26\) letters in the English alphabet. Of these, \(5\) are vowels and the rest are consonants. Thus

\begin{equation*} \size{C} = 26 - 5 = 21. \end{equation*}