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<Title><![CDATA[Lesson 6: Infinite Sets]]></Title>
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<Title><![CDATA[Infinite Sets]]></Title>
<Notes><![CDATA[Text Chapter 2   Section 6]]></Notes>
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<SlideText><![CDATA[Infinite Sets Text Chapter 2   Section 6 ]]></SlideText>
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<Title><![CDATA[Definition of an Infinite Set]]></Title>
<Notes><![CDATA[No in-class assignment problem]]></Notes>
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<SlideText><![CDATA[Definition of an Infinite Set An infinite set is a set whose elements can not be counted. An infinite set is one that has no last element. An infinite set is a set that can be placed into a one-to-one correspondence with a proper subset of itself. No in-class assignment problem ]]></SlideText>
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<Title><![CDATA[One-To-One Correspondence]]></Title>
<Notes><![CDATA[No in-class assignment problem]]></Notes>
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<SlideText><![CDATA[One-To-One Correspondence A 1-1 correspondence between two sets A and B is a rule that associates each element of set A with one and only one element of set B and vice versa. Two sets can be put into a 1-1 correspondence if they have the same cardinal number. No in-class assignment problem ]]></SlideText>
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<Title><![CDATA[Example of 1-1 Correspondence]]></Title>
<Notes><![CDATA[In=class Assignment 6   1, 2]]></Notes>
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<SlideText><![CDATA[Example of 1-1 Correspondence Let A={x, y, z} and B={5, 6, 7} n(A)=3  and n(B)=3 A 1-1 correspondence is possible. There are 6 possible 1-1 correspondences.   A = {x, y, z}              •!   •!   •!     B =  {5,   6,  7}      In=class Assignment 6   1, 2 ]]></SlideText>
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<Title><![CDATA[Contradiction?]]></Title>
<Notes><![CDATA[No in-class Assignment problem]]></Notes>
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<SlideText><![CDATA[Contradiction? A set is infinite if it can be put into a 1-1 correspondence with a proper subset. 1-1 correspondence says the sets must have the same cardinal number Proper subset says that one set must be smaller  in size than the other. No in-class Assignment problem ]]></SlideText>
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<Title><![CDATA[An Infinite Set]]></Title>
<Notes><![CDATA[No in-class assignment problem]]></Notes>
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<SlideText><![CDATA[An Infinite Set E = {2, 4, 6, 8, 10, & , 2n,& } n indicates which number in the sequence is being addressed. If n is 5 then the fifth number in the sequence is 2 times 5 or 10 2n is called the general term. F = {4, 8, 12, 16, 20, & ,4n,& } Is F a proper subset of E? No in-class assignment problem ]]></SlideText>
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<Title><![CDATA[The 1-1 Correspondence of Sets  E and F]]></Title>
<Notes><![CDATA[In=class Assignment 6  - 3]]></Notes>
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<SlideText><![CDATA[The 1-1 Correspondence of Sets   E and F E = {2, 4,  6,  8, 10, & , 2n,& }         •!   •!    •!     •!    •!           •! F = {4, 8, 12, 16, 20, & ,4n,& } The 16th element of E is 2 times 16 =32 The 16th element of F is 4 times 16 = 64 What is the 350th element of each set? Since F is a proper subset of E and there is a 1-1 correspondence shown then E is infinite. In=class Assignment 6  - 3 ]]></SlideText>
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<Title><![CDATA[Countable Sets]]></Title>
<Notes><![CDATA[In-class Assignment 6   5, 6.  First statement about cardinal numbers taken from text page 91.]]></Notes>
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<SlideText><![CDATA[Countable Sets Cantor called the cardinal number of infinite sets  transfinite cardinal numbers.  A set is countable if it is finite or if it can be placed in a 1-1 correspondence with the set of natural numbers,  N = {1, 2, 3, & }. A countable set that is infinite has a cardinality of aleph-null.  The symbol for aleph-null is      . In-class Assignment 6   5, 6.   First statement about cardinal numbers taken from text page 91. ]]></SlideText>
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