Principles of Mathematics (Math 2450)
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Chapter 5
Denumerable and Non-denumerable Sets
Dedekind’s definition of an infinite set is used to discuss properties of infinite sets and finite sets.
It is proved, among other things, that denumerable sets are the smallest in size among the infinite
sets. Properties and examples of denumerable sets and non-denumerable sets are given.
5.1 Finite and infinite sets
In Section 2.1 we defined a finite set to be a set which contains only finitely many elements. We will
give a more precise mathematical definition.
Definition. (Finite and infinite sets)
(1) A set X is said to be infinite if it has a proper subset Y such that there exists a one-to-one
correspondence between X and Y . In other words, a set X is infinite if and only if there exists
an injection f : X → X such that f (X) is a proper subset of X.
(2) A set is said to be finite if it is not infinite.
Remark. If f : X → X is a function, then we know that f (X) ⊆ X. Thus, to prove that f (X) is
a proper subset of X, we only have to prove that f (X) 6= X; that is we have to prove that there is
y ∈ X with y ∈
/ f (X).
Proposition 5.1. (N is an infinite set)
The set N of all natural numbers is an infinite set.
Proof.
1
Proposition 5.2. (φ and singleton sets are finite)
(a) The empty set φ is finite.
(b) A singleton set is finite.
Proof.
Theorem 5.1. (Subset of a finite set is finite and superset of infinite set is infinite)
(a) Every superset of an infinite set is infinite.
(b) Every subset of a finite set is finite.
Proof.
2
Proof.
Theorem 5.2. (No bijection between infinite and finite sets)
Let g : X → Y be a one-to-one correspondence. If the set X is infinite, then Y is infinite.
Proof.
3
Proof.
Corollary 5.1.
Let g : X → Y be a one-to-one correspondence. If the set X is finite, then Y is finite.
Proof.
4
Theorem 5.3. (X is infinite implies X − {x0 } is infinite)
Let X be an infinite set and let x0 ∈ X. Then X − {x0 } is infinite.
Proof.
5
Corollary 5.2. (Nm is finite)
For each m ∈ N, Nm = {1, 2, · · · , m} is finite.
Proof.
Example. Prove that if A is a set such that every proper superset of A is infinite, then A is infinite.
Solution.
Theorem 5.4. (Criterion for finite sets)
A set X is finite if and only if either X = φ or X is in one-to-one correspondence with some
Nk = {1, 2, · · · , k}.
Proof.
6
Proof.
Remark. If X is a nonempty finite set, then ∃f : Nk → X for some k ∈ N such that f is a bijection.
If we denote f (j) = xj , then X may be denoted as X = {x1 , x2 , · · · , xk }.
Proposition 5.3. (The union of two finite sets is finite)
If A and B are finite sets, then A ∪ B is finite.
Proof.
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Exercise 5.1 (3-14, 16,17)
Additional Exercises
1. Prove that A = {3n : n ∈ N} is an infinite set.
2. Prove that A = {
1
: n ∈ N} is an infinite set.
n
3. Prove that A = {2n : n ∈ Z} is an infinite set.
4. Prove that A = {n + m : n, m ∈ Q} is an infinite set.
5. Prove that X = {1, 2, 3, 5, 8, 13, 21, · · · } is an infinite set.
8
5.2 Equipotence of sets (( Numerically equivalent sets ))
Two finite sets X and Y have the same number of elements if and only if there is a bijection
f : X → Y . Although the phrase ”same number of elements” does not apply when X and Y are
infinite sets, it seems natural to think that two infinite sets that are in one-to-one correspondence
are of the same size (are numerically equivalent).
Definition. (Equipotent sets or Numerically equivalent sets )
Two sets X and Y are said to be equipotent, denoted by X ∼ Y , if there exists a bijection f : X → Y .
√
Example. Show that {1, 2, 3} and {π, 5, e} are equipotent.
Solution.
Theorem 5.5. ( ∼ is an equivalence relation)
Let S be a nonempty collection of nonempty sets and define a relation R on S by (X, Y ) ∈ R if and
only if X ∼ Y . Then R is an equivalence relation on S.
Proof. Exercise
Notation. If f : X → Y is a bijection, then we may write f : X ∼ Y .
Example 1. Show that (1, 2) ∼ (0, 2).
Solution.
π π
Example 2. Show that (− , ) ∼ R.
2 2
Solution.
9
Example 3. Prove that if (X − Y ) ∼ (Y − X), then X ∼ Y .
Solution.
Theorem 5.6. (Unions of bijections are bijections)
Let X, Y, W, and V be sets with X ∩ W = φ = Y ∩ V . If f : X ∼ Y and g : W ∼ V , then
f ∪ g : (X ∪ W ) ∼ (Y ∪ V ).
Proof.
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Remark. Theorem(5.6) also says that if X ∼ Y and W ∼ V , then (X ∪ W ) ∼ (Y ∪ V ).
Theorem 5.7. (X ∼ Y and W ∼ V implies (X × W ) ∼ (Y × V ) )
Let X, Y, W, and V be sets. If X ∼ Y and W ∼ V , then (X × W ) ∼ (Y × V ).
Proof.
Definition. (Denumerable and countable sets )
(1) A set X is said to be denumerable if X ∼ N.
(2) A set X is said to be countable if it is either finite or denumerable.
Remark. If X is a denumerable set, then ∃f : N → X such that f is a bijection. If we denote
f (j) = xj , then X may be denoted as X = {x1 , x2 , · · · }.
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Example 1. Prove that Ne = {2x : x ∈ N} is denumerable.
Solution.
Example 2. Prove that A = {1, 2, · · · , 10} is countable.
Solution.
Example 3. Prove that No = {2x − 1 : x ∈ N} is countable.
Solution.
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Proposition 5.4. (Z is denumerable )
The set Z of all integers is denumerable.
Proof.
Theorem 5.8. (Denumerable sets are the smallest infinite sets)
Every infinite subset of a denumerable set is denumerable.
Proof.
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Corollary 5.3. (A subset of a countable set is countable)
Every subset of a countable set is countable.
Proof.
Proposition 5.3. (The union of a denumerable set and a singleton set is denumerable)
If X is denumerable, then X ∪ {x0 } is denumerable for every element x0 .
Proof.
Example. Prove that if X is denumerable and Y is a finite set, then X ∪ Y is denumerable.
Solution.
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Exercise 5.2 (3-11)
Additional Exercises
1. Prove that A = {3n − 1 : n ∈ N} is a denumerable set.
2. Prove that A = {
1
: n ∈ N} is a countable set.
n
3. Prove that the set X = {2n − 1 : n ∈ N} is denumerable.
4. Prove that set A = {0, 3, 8, 15, 24, · · · } is countable.
5. Prove that set N − {3, 4} is denumerable.
6. Prove that set N − {2, 6} is denumerable.
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5.3 Examples and properties of denumerable sets
Theorem 5.9. (The union of two denumerable sets is denumerable)
If A and B are denumerable sets, then A ∪ B is denumerable.
Proof.
Corollary 5.4. (Finite union of denumerable sets is denumerable )
n
[
Let A1 , A2 , · · · , An be denumerable sets. Then
Aj is denumerable.
j=1
Proof. By induction: P (n) :
n
[
Aj is denumerable for n ≥ 2. Exercise.
j=1
Example. Show that Z is denumerable.
Solution.
16
Theorem 5.10. The set N × N is denumerable.
Proof.
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Corollary 5.5. (Infinite union of disjoint denumerable sets is denumerable)
For each k ∈ N let Ak be a denumerable set satisfying Aj ∩ Ak = φ for all j 6= k. Then
∞
[
j=1
denumerable.
Proof.
Corollary 5.6. The set Q of all rational numbers is denumerable.
Proof.
18
Aj is
Theorem 5.11. (Denumerable sets are the smallest infinite sets)
Every infinite set contains a denumerable subset.
Proof.
Exercise 5.3 (1-10)
Additional Exercises
1. Show that X = {(2x, 2y + 1) : x, y ∈ N} is denumerable.
2. Let X ⊆ R and assume that X is denumerable. Let Y = {2x : x ∈ X}. Prove that Y is
denumerable.
3. Show that Q ∩ (0, 1) is denumerable.
x
4. Show that {
2
y
3
: x, y ∈ N} is denumerable.
5. Let A be a denumerable set and let X = {{a} : a ∈ A}. Prove that X is denumerable.
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5.4 Non-denumerable sets
Lemma 5.1. Every x ∈ (0, 1) can be written as x = 0. x1 x2 · · · , where xj ∈ {0, 1, 2, · · · , 9}.
Remark. Geometric series can be used to prove that a decimal that ends with an infinite string of
9’s is equal to one that ends with an infinite string of 0’s. For example 0.31999999 · · · = 0.320000 · · · .
Definition. (Normalized form)
A decimal representation of a real number x is in normalized form if and only if there is no k such
that for all n > k, xj = 9.
Lemma 5.2. Two decimal numbers in normalized form are equal if and only if they have identical
digits in each decimal position.
Theorem 5.12. ((0, 1) is non-denumerable)
The open interval (0, 1) of real numbers is a non-denumerable set.
Proof.
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Corollary 5.7. ( R is non-denumerable)
The set R of all real numbers is non-denumerable.
Proof.
Example. Show that the set of all irrational numbers is non-denumerable.
Solution.
Exercise 5.4 (1-10)
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