Finite and Infinite Sets MATH 464/506, Real Analysis J. Robert Buchanan Department of Mathematics Summer 2007
Basic Definitions Definition The empty set has 0 elements. If n N, a set S is said to have n elements if there exists a bijection from the set to N n = {1, 2,..., n} onto S. A set S is said to be finite if it is either empty or it has n elements for some n N. A set S is said to be infinite if it is not finite.
Basic Questions Can a finite set have n elements for more than one value of n? If S is a finite set, then the number of elements in S is a unique number in N. Is the set of natural numbers finite? The set N of natural numbers is an infinite set.
Basic Questions Can a finite set have n elements for more than one value of n? If S is a finite set, then the number of elements in S is a unique number in N. Is the set of natural numbers finite? The set N of natural numbers is an infinite set.
Basic Questions Can a finite set have n elements for more than one value of n? If S is a finite set, then the number of elements in S is a unique number in N. Is the set of natural numbers finite? The set N of natural numbers is an infinite set.
Basic Questions Can a finite set have n elements for more than one value of n? If S is a finite set, then the number of elements in S is a unique number in N. Is the set of natural numbers finite? The set N of natural numbers is an infinite set.
Properties of Sets If A is a set with m elements and B is a set with n elements and if A B =, then A B has m + n elements. If A is a set with m N elements and C A is a set with 1 element, then A\C is a set with m 1 elements. If C is an infinite set and B is a finite set, then C\B is an infinite set.
Properties of Sets If A is a set with m elements and B is a set with n elements and if A B =, then A B has m + n elements. If A is a set with m N elements and C A is a set with 1 element, then A\C is a set with m 1 elements. If C is an infinite set and B is a finite set, then C\B is an infinite set. Suppose S and T are sets and that T S. If S is a finite set, then T is a finite set. If T is an infinite set, then S is an infinite set.
Countable Sets Definition A set S is said to be denumerable (or countably infinite) if there exists a bijection of N onto S. A set S is said to be countable if it is either finite or denumerable. A set S is said to be uncountable if it is not countable.
Countable Sets Definition A set S is said to be denumerable (or countably infinite) if there exists a bijection of N onto S. A set S is said to be countable if it is either finite or denumerable. A set S is said to be uncountable if it is not countable. Examples
Properties of Countable and Uncountable Sets The set N N is denumerable.
Properties of Countable and Uncountable Sets The set N N is denumerable. Suppose S and T are sets and that T S. If S is a countable set, then T is a countable set. If T is an uncountable set, then S is an uncountable set.
Properties of Countable and Uncountable Sets The set N N is denumerable. Suppose S and T are sets and that T S. If S is a countable set, then T is a countable set. If T is an uncountable set, then S is an uncountable set. The following statements are equivalent: 1 S is a countable set. 2 There exists a surjection of N onto S. 3 There exists an injection of S into N.
Rational Numbers The set Q of all rational numbers is denumerable.
Union of Countable Sets If A m is a countable set for each m N, then the union is countable. A = m=1 A m
Cantor s If A is any set, then there is no surjection of A onto the set P(A) of all subsets of A.
Cantor s If A is any set, then there is no surjection of A onto the set P(A) of all subsets of A. Remarks: There is an unending progression of larger and larger sets. P(N) is uncountable.
Homework Read Sec. 1.3. Page 21: 2, 4, 6, 8, 11 Boxed problems should be written up separately and handed in for grading at class time on Friday.