1.4 Cardinality. Tom Lewis. Fall Term Tom Lewis () 1.4 Cardinality Fall Term / 9

Transcription

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

2 Outline 1 Functions 2 Cardinality 3 Cantor s theorem Tom Lewis () 1.4 Cardinality Fall Term / 9

3 Functions Definition Let A and B be sets and let f : A B be a function. Tom Lewis () 1.4 Cardinality Fall Term / 9

4 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 (a ) implies a = a. Tom Lewis () 1.4 Cardinality Fall Term / 9

5 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 (a ) implies a = a. 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 / 9

6 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 (a ) implies a = a. 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 / 9

7 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 / 9

8 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 / 9

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. Tom Lewis () 1.4 Cardinality Fall Term / 9

10 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 / 9

11 Cardinality 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 / 9

12 Cardinality Definition A set S is Example 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. Z is denumerable. Tom Lewis () 1.4 Cardinality Fall Term / 9

13 Cantor s theorem R is not countable. Tom Lewis () 1.4 Cardinality Fall Term / 9

14 Cantor s theorem R is not countable. Corollary The intervals [a, b] and (a, b) are uncountable. Tom Lewis () 1.4 Cardinality Fall Term / 9

15 Cantor s theorem We present some additional results concerning cardinality. Tom Lewis () 1.4 Cardinality Fall Term / 9

16 Cantor s theorem We present some additional results concerning cardinality. Each infinite set S contains a denumerable subset. Tom Lewis () 1.4 Cardinality Fall Term / 9

17 Cantor s theorem We present some additional results concerning cardinality. Each infinite set S contains a denumerable subset. An infinite subset of a denumerable set is denumerable. Tom Lewis () 1.4 Cardinality Fall Term / 9

18 Cantor s theorem We present some additional results concerning cardinality. Each infinite set S contains a denumerable subset. An infinite subset of a denumerable set is denumerable. Example The set of even integers is denumerable. Tom Lewis () 1.4 Cardinality Fall Term / 9

19 Cantor s theorem N N is denumerable. Tom Lewis () 1.4 Cardinality Fall Term / 9

20 Cantor s theorem N N is denumerable. Corollary The Cartesian product of denumerable sets is denumerable. Tom Lewis () 1.4 Cardinality Fall Term / 9

21 Cantor s theorem N N is denumerable. Corollary The Cartesian product of denumerable sets is denumerable. If f : N B is a surjection and if B is infinite, then B is denumerable. Tom Lewis () 1.4 Cardinality Fall Term / 9

22 Cantor s theorem N N is denumerable. Corollary The Cartesian product of denumerable sets is denumerable. 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 / 9

23 Cantor s theorem N N is denumerable. Corollary The Cartesian product of denumerable sets is denumerable. 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 / 9