Functions: Part 4

1
Functions Part 4

• Section 7.5

2
Cardinality

• Cardinal Number
• Let S be any set
• S denotes the number of elements in the set S
• Ex 0, a 1 a, b, c 3
• Since cardinal numbers refer also to infinite
  sets, we must define what it means for two sets
  (finite or infinite) to have the same cardinal
  number (i.e., the same number of elements)
• Cardinality
• A set A has the same cardinality as a set B iff
  there is a one-to-one correspondence from A to B
  or A B
• Remember one-to-one z maps to unique y and
  onto every element in y maps to an x

3
Properties of Cardinality

• Basic Properties of Cardinality
• Reflexive A has the same cardinality as A
• Symmetric If A has the same cardinality as B,
  then B has the same cardinality as A
• Transitive If A has the same cardinality as B,
  and B has the same cardinality as C, then A has
  the same cardinality as C
• Countability
• A set S is finite iff it is empty, or if there is
  a natural number n such that S and 1, 2, 3, ,
  n have the same cardinality
• A set is infinite if it is not finite
• A set S is countably infinite iff S Z
• A set S is countable iff it is finite or
  countably infinite
• A set is uncountable iff it is not countable

4
Countability

• The set of positive integers Z 1, 2, 3, is
  an infinite set
• A set B having the same cardinality as Z is
  called countably infinite because the one-to-one
  correspondence between the two sets can be used
  to count the elements of B

Important Characteristic of Infinite Sets it can
have the same cardinality as a proper subset of
itself, e.g. Z Z
5
Countability Example

• Show that the set Z of all integers is countable
• We need to find a function from Z to Z that is
  one-to-one and onto

Start in the middle and work your way out
6
Example (contd)

• A function from Z to Z that is one-to-one and
  onto is
• f(n) n/2 n is an even integer
• -(n-1)/2 n is an odd integer
• A (Z) B (Z)
• n1 f(n) 0
• n2 f(n) 1
• n3 f(n) -1
• n4 f(n) 2
• n5 f(n) -2

No integer is counted twice so f(n) is one-to-one
Every element in Z (co-domain) is counted
eventually, so f(n) is onto ? f(n) has a
one-to-one correspondence from Z to Z
7
Final Thoughts on Example

• Even though there seems to be more integers Z
  than positive integers Z, the elements of the
  two sets can be paired up one-for-one
• ? The sets have the same cardinality and Z is
  countably infinite

8
Countability Proof

• Show that the set 2Z of all even integers is
  countable
• Define h Z ? 2Z such that h(n) 2n
• We know that h(n) is one-to-one and onto, hence Z
  has the same cardinality as 2Z
• From previous example, Z has the same cardinality
  as Z, so by transitivity property of cardinality
• 2Z Z and Z Z ? 2Z Z
• Therefore, 2Z is countably infinite and thus
  countable

9
Uncountable (Informal Proof)

• The set of all real numbers between 0 and 1 is
  uncountable

Any interval of real numbers can be divided into
10 equal subintervals, then the process of
obtaining additional digits in the decimal
expansion can (theoretically) be repeated
indefinitely
10
Cardinality of the Set of all Real Numbers

• Show that the set of all real numbers has the
  same cardinality as the set of real numbers
  between 0 and 1
• S is the open interval of real numbers between 0
  and 1 S x?R 0 lt x lt 1

Take S and bend it into a circle
11
Example (contd)
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

• For each point x on the circle representing S,
  draw a straight line L thru the top most point of
  the circle
• F(x) is the intersection of L and the number line
• Distinct point on circle goes to distinct point
  on number line, so F(x) is one-to-one.
• Given any y, a line can be drawn through y to the
  top of the circle, thus y F(x) which is onto
• Therefore, there is a one-to-one correspondence