3.2: Countable and Uncountable Sets

Definition

A function \(\varphi: A \rightarrow B\) is said to be a one-to-one correspondence if \(\varphi\) is both one-to-one and onto.

Definition

We say sets \(A\) and \(B\) have the same cardinality if there exists a one-to-one correspondence \(\varphi: A \rightarrow B\).

We denote the fact \(A\) and \(B\) have the same cardinality by writing \(|A|=|B| .\)

Exercise \(\PageIndex{1}\)

Define a relation on sets by setting \(A \sim B\) if and only if \(|A|=|B| .\) Show that this relation is an equivalence relation.

Definition

Let \(A\) be a set. If, for \(n \in \mathbb{Z}^{+}, A\) has the cardinality of the set \(\{1,2,3, \ldots, n\},\) we say \(A\) is finite and write \(|A|=n .\) If \(A\) has the cardinality of \(\mathbb{Z}^{+},\) we say \(A\) is countable and write \(|A|=\aleph_{0} .\)

Example \(\PageIndex{1}\)

If we define \(\varphi: \mathbb{Z}^{+} \rightarrow \mathbb{Z}\) by

\[\varphi(n)=\left\{\begin{array}{ll}{\frac{n-1}{2},} & {\text { if } n \text { is odd, }} \\ {-\frac{n}{2},} & {\text { if } n \text { is even, }}\end{array}\right.\]

then \(\varphi\) is a one-to-one correspondence. Thus \(|\mathbb{Z}|=\aleph_{0}\).

Exercise \(\PageIndex{2}\)

Let \(A\) be the set of even integers. Show that \(|A|=\aleph_{0}\).

Exercise \(\PageIndex{3}\)

Verify each of the following:

a. If \(A\) is a nonempty subset of \(\mathbb{Z}^{+},\) then \(A\) is either finite or countable.

b. If \(A\) is a nonempty subset of a countable set \(B,\) then \(A\) is either finite or countable.

Proposition \(\PageIndex{1}\)

Suppose \(A\) and \(B\) are countable sets. Then the set \(C=\) \(A \cup B\) is countable.

Proof

Suppose \(A\) and \(B\) are disjoint, that is, \(A \cap B=\emptyset .\) Let \(\varphi: \mathbb{Z}^{+} \rightarrow A\) and \(\psi: \mathbb{Z}^{+} \rightarrow B\) be one-to-one correspondences. Define \(\tau: \mathbb{Z}^{+} \rightarrow C\) by

\[\tau(n)=\left\{\begin{array}{ll}{\varphi\left(\frac{n+1}{2}\right),} & {\text { if } n \text { is odd, }} \\ {\psi\left(\frac{n}{2}\right),} & {\text { if } n \text { is even. }}\end{array}\right.\]

Then \(\tau\) is a one-to-one correspondence, showing that \(C\) is countable.

If \(A\) and \(B\) are not disjoint, then \(\tau\) is onto but not one-to-one. However, in that case \(C\) has the cardinality of an infinite subset of \(\mathbb{Z}^{+},\) and so is countable. \(\quad\) Q.E.D.

Definition

A nonempty set which is not finite is said to be infinite. An infinite set which is not countable is said to be uncountable.

Exercise \(\PageIndex{4}\)

Suppose \(A\) is uncountable and \(B \subset A\) is countable. Show that \(A \backslash B\) is uncountable.

Proposition \(\PageIndex{2}\)

Suppose \(A\) and \(B\) are countable. Then \(C=A \times B\) is countable.

Proof

Let \(\varphi: \mathbb{Z}^{+} \rightarrow A\) and \(\psi: \mathbb{Z}^{+} \rightarrow B\) be one-to-one correspondences. Let \(a_{i}=\varphi(i)\) and \(b_{i}=\psi(i) .\) Define \(\tau: \mathbb{Z}^{+} \rightarrow C\) by letting

\[\tau(1)=\left(a_{1}, b_{1}\right),\]

\[\tau(2)=\left(a_{1}, b_{2}\right),\]

\[\tau(3)=\left(a_{2}, b_{1}\right),\]

\[\tau(4)=\left(a_{1}, b_{3}\right),\]

\[\tau(5)=\left(a_{2}, b_{2}\right),\]

\[\tau(6)=\left(a_{3}, b_{1}\right),\]

\[\tau(7)=\left(a_{1}, b_{4}\right),\]

\[\vdots = \vdots\]

That is, form the infinite matrix with \(\left(a_{i}, b_{j}\right)\) in the \(i\)th row and \(j\)th column, and then count the entries by reading down the diagonals from right to left. Then \(\tau\) is a one-to-one correspondence and \(C\) is countable.

Proposition \(\PageIndex{3}\)

\(\mathbb{Q}\) is countable.

Proof

By the previous proposition, \(\mathbb{Z} \times \mathbb{Z}\) is countable. Let

\[A=\{(p, q): p, q \in \mathbb{Z}, q>0, p \text { and } q \text { relatively prime }.\}\]

Then \(A\) is infinite and \(A \subset \mathbb{Z} \times \mathbb{Z},\) so \(A\) is countable. But clearly \(|\mathbb{Q}|=|A|,\) so \(\mathbb{Q}\) is countable. \(\quad\) Q.E.D.

Proposition \(\PageIndex{4}\)

Suppose for each \(i \in \mathbb{Z}^{+}, A_{i}\) is countable. Then

\[B=\bigcup_{i=1}^{\infty} A_{i}\]

is countable.

Proof

Suppose the sets \(A_{i}, i \in \mathbb{Z}^{+},\) are pairwise disjoint, that is, \(A_{i} \cap A_{j}=\emptyset\) for all \(i, j \in \mathbb{Z}^{+} .\) For each \(i \in \mathbb{Z}^{+},\) let \(\varphi_{i}: \mathbb{Z}^{+} \rightarrow A_{i}\) be a one-to-one correspondence. Then \(\psi: \mathbb{Z}^{+} \times \mathbb{Z}^{+} \rightarrow B\) defined by

\[\psi(i, j)=\varphi_{i}(j)\]

is a one-to-one correspondence, and so \(|B|=\left|\mathbb{Z}^{+} \times \mathbb{Z}^{+}\right|=\aleph_{0}\).

If the sets \(A_{i}, i \in \mathbb{Z}^{+},\) are not disjoint, then \(\psi\) is onto but not one-to-one. But then there exists a subset \(P\) of \(\mathbb{Z}^{+} \times \mathbb{Z}^{+}\) such that \(\psi: P \rightarrow B\) is a one-to-one correspondence. Since \(P\) is an infinite subset of a countable set, \(P\) is countable and so \(|B|=\aleph_{0} .\) \(\quad\) Q.E.D.

If in the previous proposition we allow that, for each \(i \in \mathbb{Z}^{+}, A_{i}\) is either finite or countable, then \(B=\bigcup_{i=1}^{\infty} A_{i}\) will be either finite or countable.