Fall 2015 COMP 2300 Discrete Structures for Computation Donghyun (David) Kim Department of Mathematics and Physics North Carolina Central University 1 Chapter 7.4 Cardinality of Sets

Fall 2015 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University Finite and Infinite Sets Historically, the term cardinal number was introduced to describe the size of a set. A finite set is one that has no element at all or that can be put into one-to-one correspondence with {1, 2, …, n } for some positive integer n. An infinite set is a nonempty set that cannot be put into one-to-one correspondence with {1, 2, …, n } for any positive integer n. 2

Fall 2015 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University Cardinality of Sets Let A and B by any sets. A has the same cardinality as B if, and only if, there is an one-to-one correspondence from A to B. In other words, A has the same cardinality as B if, and only if, there is a function f from A to B that is one-to- one and onto. 3

Fall 2015 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University Properties of Cardinality For all sets A, B, and C : a.Reflexive property of cardinality: A has the same cardinality as A. b.Symmetric property of cardinality: If A has the same cardinality as B, then B has the same cardinality as A. c.Transitive property of cardinality: If A has the same cardinality as B and B has the same cardinality as C, then A has the same cardinality as C. By b, A and B have the same cardinality if, and only if, A has the same cardinality as B or B has the same cardinality as A. 4

Fall 2015 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University Infinite Sets and Proper Subsets with the Same Cardinality Q) Let 2Z be the set of all even integers. Prove that 2Z and Z have the same cardinality. A) Consider the function defined as follows: Now, to prove the theorem, we need to show that 1.H is one-to-one, and 2.H is onto. 5

Fall 2015 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University Infinite Sets and Proper Subsets with the Same Cardinality Q) Let 2Z be the set of all even integers. Prove that 2Z and Z have the same cardinality. A) Consider the function defined as follows: 1.H is one-to-one 6

Fall 2015 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University Infinite Sets and Proper Subsets with the Same Cardinality Q) Let 2Z be the set of all even integers. Prove that 2Z and Z have the same cardinality. A) Consider the function defined as follows: 2. H is onto. 7

Fall 2015 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University Countable Sets 8 The set of counting numbers {1, 2, 3, 4, …} is, in a sense, the most basic of all infinite sets. A set A having the same cardinality as this set is called countably infinite. A set is called countable if and only if, it is finite or countably infinite. A set that is not countable is called uncountable.

Fall 2015 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University Countability of Z, the Set of All Integers 9 Show that the set Z of all integers is countable. 0

Fall 2015 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University Countability of Z, the Set of All Integers 10 Show that the set Z of all integers is countable. 1

Fall 2015 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University Countability of Z, the Set of All Integers 11 Show that the set Z of all integers is countable. 2

Fall 2015 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University Countability of Q, the Set of All Positive Rational Numbers 12 Show that the set is countable.

Fall 2015 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University Other Theorems 13 The set of real numbers, R, is uncountable. Proof (by contradiction) Suppose there is an one-to-one correspondence f which sends the set of real numbers in [0,1] to {1, …, n } for some n. Then, given any fixed number n is, you can find a set of real numbers C in [0,1] such that | C | > n. No one-to-one function can be defined over the domain with size larger than n and co-domain with size n. Any subset of any countable set is countable. Corollary: Any set with an uncountable subset is uncountable. (contrapositvie)

Fall 2015 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University The Cardinality of the Set of All Real Numbers 14 Show that the set of all real numbers has the same cardinality as the set of real numbers between 0 and 1.

Fall 2015 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University The Cardinality of the Set of All Real Numbers 15 Show that the set of all real numbers has the same cardinality as the set of real numbers between 0 and 1.