ee1j2 - slide 1 ee1j2 – discrete maths lecture 9 isomorphic sets cardinality countability and ...

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  • Slide 1
  • EE1J2 - Slide 1 EE1J2 Discrete Maths Lecture 9 Isomorphic sets Cardinality Countability and 0 Countability of and
  • Slide 2
  • EE1J2 - Slide 2 Isomorphic Sets If there exists a bijection between two sets A and B, f: A B, then: The elements of A and B are in one-to-one correspondence A and B are basically the same set A and B are isomorphic f 1-1 and onto - bijection
  • Slide 3
  • EE1J2 - Slide 3 Example 1 Let A = {0,1,2,3} and B = {a,b,c,d} The function f :A B defined by {(0,a),(1,b),(2,c),(3,d)} is a bijection The sets A and B are isomorphic. B is just a re-labelled version of A
  • Slide 4
  • EE1J2 - Slide 4 Example 2 Recall that is the set of positive whole numbers and is the set of all whole numbers. Then the function: f : , f(n) = n/2 if n is even, f(n)=-(n+1)/2 if n is odd is a bijection Hence and are isomorphic
  • Slide 5
  • EE1J2 - Slide 5 Example 2 continued f : , f(n) = n/2 if n is even, f(n)=-(n+1)/2 if n is odd 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 . . 0 -1 1 -2 2 -3 3 -4 4 -5 5 -6 6 -7 7 -8 8 .
  • Slide 6
  • EE1J2 - Slide 6 Cardinality Revisited Recall that for a finite set A={a 1,,a n }, the cardinality of A is simply the number of members which A has. In this case |A|=n For infinite sets the notion of cardinality is more complex. But, if two infinite sets A and B are isomorphic, then surely |A|=|B|
  • Slide 7
  • EE1J2 - Slide 7 Cardinality Revisited In other words, if we can find a bijection f:A B, then |A|=|B| Because, in this case B is just A with different labels
  • Slide 8
  • EE1J2 - Slide 8 Countable Sets The simplest infinite set is the set of natural numbers = {0,1,2,3,} E.g: if n then we can talk about the next biggest member of , i.e. n+1 The same is not true of the real numbers or even the rational numbers Given any n , we also know that by counting through the numbers, starting at 0, we will eventually reach n. We can count
  • Slide 9
  • EE1J2 - Slide 9 Countability A set A is called countable if it is isomorphic with i.e A is countable if there exists a bijection f: A The cardinality of , | |, is denoted by 0 pronounced aleph zero
  • Slide 10
  • EE1J2 - Slide 10 Countability of It is easy to show that the set of integers is countable. Remember, = {,-3,-2,-1,0,1,2,3,} Define f: (as in previous example) by:
  • Slide 11
  • EE1J2 - Slide 11 Countability of Claim: f is a bijection f is 1-1 Suppose f(m)=f(n). If f(m)=f(n) is positive, then m=2f(m)=2f(n)=n. If f(m)=f(n) is negative, then m=2f(m)-1=2f(n)-1=n f is onto, by definition Hence f is a bijection So, is countable, and | |= 0
  • Slide 12
  • EE1J2 - Slide 12 Countability and listing Sometimes it is difficult to write down the bijection f : A which shows that A is countable The first step in constructing a bijection with is to show that A can be written as a list Take the integers in the previous example: 0 -1 1 -2 2 -3 3 -4 4 -5 5 -6 6 -7 7 -8 8 . 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 . .
  • Slide 13
  • EE1J2 - Slide 13 Countability of Recall that is the set {(n,m) : n,m } Claim: is countable How can we write as a list? (0,1) (0,2) (0,3) (0,n) ? This is no good well never get to (1,1) for example
  • Slide 14
  • EE1J2 - Slide 14 Countability of 0 1 2 3 4 5 6 . 0 (0,0) (0,1) (0,2) (0,3) (0,4) (0,5) (0,6) 1 (1,0) (1,1) (1,2) (1,3) (1,4) (1,5) (1,6) 2 (2,0) (2,1) (2,2) (2,3) (2,4) (2,5) (2,6) 3 (3,0) (3,1) (3,2) (3,3) (3,4) (3,5) (3,6) 4 (4,0) (4,1) (4,2) (4,3) (4,4) (4,5) (4,6) :
  • Slide 15
  • EE1J2 - Slide 15 Countability of So the countability of is demonstrated by the list: (0,0) (1,0) (1,1) (2,0) (2,1) (2,2) (3,0) (3,1) (3,2) All pairs containing just 0 All pairs containing just 0 and 1 and not in red section All pairs containing just 0, 1 or 2 and not in red or blue sections
  • Slide 16
  • EE1J2 - Slide 16 Countability of It is not surprising that is countable or that is countable A much more surprising result, demonstrated by the mathematician George Cantor, is that is countable This is counter-intuitive, but it is true. In other words, in some sense there are no more rational numbers than there are natural numbers
  • Slide 17
  • EE1J2 - Slide 17 Cantors Proof of the Countability of Cantors proof is based on a particular ordering of For each n let S n be the set of rational numbers x=a/b such that n=max{|a|,b}and such that x S m for m