§1.4–Cardinality

Tom Lewis

Fall Term 2006

Tom Lewis () §1.4–Cardinality Fall Term 2006 1 / 9 Outline

1 Functions

2 Cardinality

3 Cantor’s theorem

Tom Lewis () §1.4–Cardinality Fall Term 2006 2 / 9 f is said to be an injection (or one-to-one) provided that

f (a) = f (a0) implies a = a0.

f is said to be a surjection (or onto) provided that for each b ∈ B there exists an a ∈ A such that f (a) = b. f is said to be a bijection provided that f is an injection and a surjection, that is, f is one-to-one and onto.

Functions

Definition Let A and B be sets and let f : A → B be a function.

Tom Lewis () §1.4–Cardinality Fall Term 2006 3 / 9 f is said to be a surjection (or onto) provided that for each b ∈ B there exists an a ∈ A such that f (a) = b. f is said to be a bijection provided that f is an injection and a surjection, that is, f is one-to-one and onto.

Functions

Definition Let A and B be sets and let f : A → B be a function. f is said to be an injection (or one-to-one) provided that

f (a) = f (a0) implies a = a0.

Tom Lewis () §1.4–Cardinality Fall Term 2006 3 / 9 f is said to be a bijection provided that f is an injection and a surjection, that is, f is one-to-one and onto.

Functions

Definition Let A and B be sets and let f : A → B be a function. f is said to be an injection (or one-to-one) provided that

f (a) = f (a0) implies a = a0.

f is said to be a surjection (or onto) provided that for each b ∈ B there exists an a ∈ A such that f (a) = b.

Tom Lewis () §1.4–Cardinality Fall Term 2006 3 / 9 Functions

Definition Let A and B be sets and let f : A → B be a function. f is said to be an injection (or one-to-one) provided that

f (a) = f (a0) implies a = a0.

f is said to be a surjection (or onto) provided that for each b ∈ B there exists an a ∈ A such that f (a) = b. f is said to be a bijection provided that f is an injection and a surjection, that is, f is one-to-one and onto.

Tom Lewis () §1.4–Cardinality Fall Term 2006 3 / 9 Show that Z ∼ N. Show that a set of 10 elements cannot have the same cardinality as a set of 11 elements.

Problem

Cardinality

Definition The sets A and B are said to have the same cardinality, written A ∼ B, provided that there exists a bijection f : A → B. We write

cardA = cardB if A and B have the same cardinality.

Tom Lewis () §1.4–Cardinality Fall Term 2006 4 / 9 Show that Z ∼ N. Show that a set of 10 elements cannot have the same cardinality as a set of 11 elements.

Cardinality

Definition The sets A and B are said to have the same cardinality, written A ∼ B, provided that there exists a bijection f : A → B. We write

cardA = cardB if A and B have the same cardinality.

Problem

Tom Lewis () §1.4–Cardinality Fall Term 2006 4 / 9 Show that a set of 10 elements cannot have the same cardinality as a set of 11 elements.

Cardinality

Definition The sets A and B are said to have the same cardinality, written A ∼ B, provided that there exists a bijection f : A → B. We write

cardA = cardB if A and B have the same cardinality.

Problem Show that Z ∼ N.

Tom Lewis () §1.4–Cardinality Fall Term 2006 4 / 9 Cardinality

Definition The sets A and B are said to have the same cardinality, written A ∼ B, provided that there exists a bijection f : A → B. We write

cardA = cardB if A and B have the same cardinality.

Problem Show that Z ∼ N. Show that a set of 10 elements cannot have the same cardinality as a set of 11 elements.

Tom Lewis () §1.4–Cardinality Fall Term 2006 4 / 9 Cardinality

Theorem The relation ∼ is an equivalence relation: A ∼ A A ∼ B implies B ∼ A A ∼ B and B ∼ C implies A ∼ C.

Tom Lewis () §1.4–Cardinality Fall Term 2006 5 / 9 Cardinality

Definition A set S is finite if it is empty or for some n ∈ N, S ∼ {1, 2,..., n}; infinite if it is not finite; denumerable if S ∼ N; countable if it is finite or denumerable; uncountable if it is not countable.

Example Z is denumerable.

Tom Lewis () §1.4–Cardinality Fall Term 2006 6 / 9 Corollary The intervals [a, b] and (a, b) are uncountable.

Cantor’s theorem

Theorem R is not countable.

Tom Lewis () §1.4–Cardinality Fall Term 2006 7 / 9 Cantor’s theorem

Theorem R is not countable.

Corollary The intervals [a, b] and (a, b) are uncountable.

Tom Lewis () §1.4–Cardinality Fall Term 2006 7 / 9 Theorem Each infinite set S contains a denumerable subset.

Theorem An infinite subset of a denumerable set is denumerable.

Example The set of even integers is denumerable.

Cantor’s theorem

We present some additional results concerning cardinality.

Tom Lewis () §1.4–Cardinality Fall Term 2006 8 / 9 Theorem An infinite subset of a denumerable set is denumerable.

Example The set of even integers is denumerable.

Cantor’s theorem

We present some additional results concerning cardinality.

Theorem Each infinite set S contains a denumerable subset.

Tom Lewis () §1.4–Cardinality Fall Term 2006 8 / 9 Example The set of even integers is denumerable.

Cantor’s theorem

We present some additional results concerning cardinality.

Theorem Each infinite set S contains a denumerable subset.

Theorem An infinite subset of a denumerable set is denumerable.

Tom Lewis () §1.4–Cardinality Fall Term 2006 8 / 9 Cantor’s theorem

We present some additional results concerning cardinality.

Theorem Each infinite set S contains a denumerable subset.

Theorem An infinite subset of a denumerable set is denumerable.

Example The set of even integers is denumerable.

Tom Lewis () §1.4–Cardinality Fall Term 2006 8 / 9 Corollary The Cartesian product of denumerable sets is denumerable.

Theorem If f : N → B is a surjection and if B is infinite, then B is denumerable.

Corollary The denumerable union of denumerable sets is denumerable.

Corollary Q is denumerable.

Cantor’s theorem

Theorem N × N is denumerable.

Tom Lewis () §1.4–Cardinality Fall Term 2006 9 / 9 Theorem If f : N → B is a surjection and if B is infinite, then B is denumerable.

Corollary The denumerable union of denumerable sets is denumerable.

Corollary Q is denumerable.

Cantor’s theorem

Theorem N × N is denumerable.

Corollary The Cartesian product of denumerable sets is denumerable.

Tom Lewis () §1.4–Cardinality Fall Term 2006 9 / 9 Corollary The denumerable union of denumerable sets is denumerable.

Corollary Q is denumerable.

Cantor’s theorem

Theorem N × N is denumerable.

Corollary The Cartesian product of denumerable sets is denumerable.

Theorem If f : N → B is a surjection and if B is infinite, then B is denumerable.

Tom Lewis () §1.4–Cardinality Fall Term 2006 9 / 9 Corollary Q is denumerable.

Cantor’s theorem

Theorem N × N is denumerable.

Corollary The Cartesian product of denumerable sets is denumerable.

Theorem If f : N → B is a surjection and if B is infinite, then B is denumerable.

Corollary The denumerable union of denumerable sets is denumerable.

Tom Lewis () §1.4–Cardinality Fall Term 2006 9 / 9 Cantor’s theorem

Theorem N × N is denumerable.

Corollary The Cartesian product of denumerable sets is denumerable.

Theorem If f : N → B is a surjection and if B is infinite, then B is denumerable.

Corollary The denumerable union of denumerable sets is denumerable.

Corollary Q is denumerable.

Tom Lewis () §1.4–Cardinality Fall Term 2006 9 / 9