Sets

1
Sets
2
Preliminaries

• Set is a primitive concept in mathematics It
  is not defined.
• Visualize a dictionary as a directed graph.
• The nodes represent words
• If word w is defined in terms of word u, then
  there is an edge from w to u.
• Cycles? Finite?
• Intuitively, a set is a collection of elements.

3
Preliminaries

• The universe of discourse is a set describing the
  context for the duration of the discussion.
• For example, the set of all humans.
• The universe (for short) must be explicitly
  stated the truth value of a statement depends on
  it (as we shall see)
• But since the universe is itself a set, it too
  must have a universe, ?

4
Preliminaries

• A set S is well-defined when we can decide
  whether any particular object in the universe of
  discourse is an element of S.
• S is the set of all even numbers
• S is the set of all human beings
• Do we all agree on whether some blob of
  protoplasm is a human being?

5
Notation

• Sets typically are denoted with an upper-case
  letter.
• Elements typically are denoted with a lower-case
  letter.
• (Why do I say typically?)
• Order of presentation is irrelevant
• a, b, c c, b, a

6
Operators

• I assume that you know the definition of the
  standard operators
• ?, ?, ?, ?, ?, ?.
• (Illustrate ? with a Venn diagram.)
• The cardinality (aka size) of a set is the number
  of elements in it.
• In this course, we deal primarily with finite
  sets.
• The size of a set S is denoted S.

7
Proving a set equation

• Using a Venn diagram (draw both sides)

8
Proving a set equation

• From the definition of set operators

9
Proving a set equation by identities
10
Disproving a set equation

• means for all sets A, B, C the equation holds.
• Proving an equation is false means
• it is not the case that for all A, B, C,
• That is, there exists sets A, B, C

11
Disproving a set equation

• Constructively A B 1, C ?.
• We have a counterexample.
• In this case, Venn diagrams are instructive.
• Proving that A B is false is not the same as
  proving A ? B. Explain?

12
Power set

• The power set of A, denoted
• is the set of all subsets of A
• Example Let A a, b.
• P(A) ?, a, b, a,b
• If A n, then P(A) ?

13
If A n then P(A) 2A

• Let A a1, a2, . . . , an.
• Associate any S ? A with an n-bit string B as
  follows
• ai ? S if and only if the ith bit of B is 1.
• For A a1, a2
• subset ? corresponds to string 00
• subset a1 corresponds to string 10
• subset a2 corresponds to string 01
• subset a1, a2 corresponds to string 11

14
Proving A B

• Each subset corresponds to a string
• Each string corresponds to a subset
• Therefore the subsets are in a one-to-one
  correspondence with the set of strings.
• Since there are 2n n-bit strings, if A n,
  then P(A) 2n.
• Visualize the elements of P(A) arranged as a
  hypercube.

15
Proving A B

• Are there more natural numbers than integers?
• Are there more rational numbers than natural
  numbers?
• Are there more points in this line ____ than in
  this line ____________ ?

16
The cardinality of sets

• A lt P(A) is a theorem of set theory.
• Let N denote the set of natural numbers.
• N lt P(N).
• Let C the set of functions f N ? 0, 1.
• Are all such functions computable?
• That is, for each such function, is there a
  computer program that computes it?