Lesson 5: Sets

1
Lesson 5 Sets

• Objectives
• Identify a set and use set notation
• Use set builder notation and translate such
  notation to the set it represents.
• Represent Sets with Venn Diagrams
• Use the terminology in Section 1.6 to discuss and
  describe sets
• Give the Power set of a set S
• Use the set identities on pg 89
• Prove equivalencies using Venn Diagrams and
  identities

• Outline
• Sets
• Venn Diagrams
• Terminology
• Power Sets
• Cartesian Products
• Identities
• Reading Section 1.6, 1.7
• Homework Due Feb 8
• 1.6 8efg, 19, 20, 24a, 28ab
• 1.7 2,4,16,17ab,21cd,40,41, 42, 49

2
Important Sets

• Natural Numbers
• Integers
• Positive Integers
• Rational Numbers
• Real Numbers

3
Set Equality

• Equality of sets is biconditional on the two sets
  having exactly the same elements
• Ordering not important
• Repetition is redundant
• A all Illini fans
• B intelligent sentient beings

4
Set Builder Notation

• O x x is even
• P x x is prime
• V x x a2/b2, a,b ? integers

5
Venn Diagrams

• Rectangle Represents Universe
• Sets represented graphically

6
Set Notation and Terminology

• Empty set ?
• also denoted
• Singleton set ? (This is a set containing
  a set!)
• Subset ?
• A ? B iff every element in A is also in B
• Odds ? Integers
• The null set is a subset of any set
• A set is a subset of itself
• The set of ECE357 students with antennae is a
  subset of ECE357 students
• The set of ECE357 students is a subset of the set
  of ECE357 students

7
Set Notation and Terminology

• If A ? B, and A ? B
• A ? B
• A is a proper subset of B
• To show A B, it is sufficient to show
• A ? B
• B ? A
• ?x(x?A ? x?B)
• Cardinality of set A is number of elements in A

8
Power Set

• The power set is the set of all subsets of a set
  S.
• S H, A, M
• T ?
• U ?

9
Tuples

• An n-tuple is an ordered collection of elements
• (Sets are unordered and unique)
• Cartesian Product (on sets A,B)
• Denoted A X B
• Set of ordered pairs (a,b)
• Matchings of all members from each set
• Extends to multiple sets

10
Cartesian Product

• A Mary, Sue, Jane, Lisa
• B Ed, Lewis, Goah
• Form ( , ), ( , ) ( , )
• Is A X B equal to B X A? Why or why not?

11
Multiple Cartesian Products

• A Bill, Ed, Jon
• B Sue, Ann
• C Unchained Melody (U.C.), Nowhere Else
  (N.E.)
• A X B X C

12
Union, Intersection, Negation

• Use Venn Diagrams to visualize
• Two sets are DISJOINT if their intersection is
  the empty set