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Chapter 5

Denumerable and Nondenumerable Sets

5.1 Finite and Infinite Sets

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) X; that is we have to

prove that there is y X with y f(X).

Proposition. (N is an infinite set)

The set N of all natural numbers is an infinite set.

Proof.

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Proposition. ( and singleton sets are finite)

(a) The empty set is finite.

Proof.

(b) A singleton set is finite.

Proof.

Theorem. (A subset of a finite set is finite and a 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.(b)

Theorem . (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.

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Corollary. Let g : X Y be a one-to-one

correspondence. If the set X is finite, then Y is finite.

Proof.

Theorem. ( X is infinite implies X – { x0 } is infinite)

Let X be an infinite set and let x0 X. Then X – { x0 } is

infinite.

Corollary. (Nk is finite)

For each k N, Nk = { 1 , 2 , … , k } is finite.

Proof.

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Example. Prove that if A is a set such that every proper

superset of A is infinite, then A is infinite.

Proof.

Theorem. (Criterion for finite sets)

A set X is finite if and only if either X = or X is in one-

to-one correspondence with some Nk = { 1 , 2 , … , k }.

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. (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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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 .

Notation. If f : X Y is a bijection, then we may write

f : X Y .

Example. Show that X = { 1 , 2 , 3 } and Y = { # , $ ,* }

are equipotent.

Proof.

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Theorem. ( 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.

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Examples. Show that

a) (0 , 1) (-1, 1).

b) (0 , 1) ( a , b) for any two real numbers a and b.

c) (0 , 1) R.

d) (0 , 1] ( - , 0]

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e) (0 , 1] [0 , ).

f) (0 , 1] [ 1 , ).

Example. Prove that if (X - Y ) (Y - X), then X Y .

Proof.

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Theorem. (Unions of bijections are bijections)

Let X , Y , W, and V be sets with X W = = Y V . If

f : X Y and g : W V , then f g : (X W) (Y V).

Theorem. (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.

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Remarks. - The smallest infinite set is N, or any set that is

equipotent to N.

- Not all infinite sets are equipotent to N.

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 , x3 ,… }.

Example. Prove that X = { 1 , 2 ,3 ,…,10 }is countable.

proof.

Example. Prove that the set of odd positive integer

is countable.

proof.

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Theorem. (Denumerable sets are the smallest infinite sets)

Every infinite subset of a denumerable set is

denumerable.

Corollary. (A subset of a countable set is countable)

Every subset of a countable set is countable.

Proof.

Proposition. (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.

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Example. Prove that if X is denumerable and Y is a finite

set, then X Y is denumerable.

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5.3 Examples and Properties of Denumerable Sets

Theorem. (The union of two denumerable sets is

denumerable)

If X and Y are denumerable sets, then X Y is

denumerable.

Corollary. (Finite union of denumerable sets is

denumerable )

Example. Show that Z is denumerable.

Proof.

Theorem. The set N N is denumerable.

Corollary. (Infinite union of disjoint denumerable sets is

denumerable)

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Corollary. The set Q of all rational numbers is

denumerable.

Theorem. (Denumerable sets are the smallest infinite

sets)

Every infinite set contains a denumerable subset.

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5.4 Nondenumerable Sets

Definition. (Nondenumerable and uncountable sets)

A set X is said to be nondenumerable or uncountable if it

is not countable.

Theorem. ((0, 1) is nondenumerable)

The open interval (0 , 1) of real numbers is a non-

denumerable set.

Corollary. ( R is nondenumerable)

The set R of all real numbers is non-denumerable.

Proof.

Example. Show that the set of all irrational numbers is

nondenumerable.

Proof.