Sets Set Operations

1
Sets Set Operations
2
Learning Objectives

• Understand what are sets, how to build them, and
  some important properties and theorems about
  sets.
• Understand which set operators exist that permit
  to create new sets.

3
Sets

• A 1, 2, ab, ba, 1, 2, 3, isabelle, table,
  1,2, ab, ba, isabelle table are elements they
  are members of the set A or belong to A.
• Notation a ? A.
• V a, i, o, u, e Set of Vowels
• O 1,3,5,7,9 Odd numbers lt 10.
• A B if x ? A ? x ? B A 1,3,5,2,1,3,5 B
  1,2,35, A B.Proving set equality
  either x ? A ? x ? B or A ? B ? B ? A

4
Sets

• Set Builder B x P(x) B1 x x n2
  1, 0 ? n ? 10 A1 B1
• Venn Diagrams.
• Subset proper subset. A ? B A ? B A
  ? B A ? B
• The empty set ? ? B
• Set may have other sets as members A ?,
  a, b, a,b. Note A has 4 elements. ? ? A
  and also ? ? B. a ? A.

5
Sets

• A is finite if it has n elements, otherwise it is
  infinite.
• Power set P(A) B B ? A.
• The power set of A contains 2n elements where n
  is the number of elements in A, also named the
  cardinality of A.
• Æ has 0 elements. P(?) has one element P(?)
  ?
• A a P(A) ?, a P(?) ?, ?

6
Set Operations

• Propositional calculus and set theory are both
  instances of an algebraic system called a
• Boolean Algebra.
• The operators in set theory are defined in terms
  of the corresponding operator in propositional
  calculus
• As always there must be a universe U. All sets
  are assumed to be subsets of U

7
Set Operations

• Cartesian product A x B (a,b) a ? A ? b ?
  B
• Can be defined using sets only A x B a,
  a,b a ? A ? b ? B
• Note (a,b) ? (b,a) if a ? b.
• Cartesian product of n sets A1 x A2 x x An
  (a1, a2, , an) ai ? Ai, i 1,,n
• Union A ? B x x ? A ? x ? B
• Intersection A ? B x x ? A ? x ? B
• Venn diagrams, use Mathematica / Maple to build
  sets.

8
Set Operations

• Maple examples for propositional logics

9
Set Operations

• Maple examples for sets

10
Set Operations

• Maple examples for sets

11
Set Operations

• Definitions
• The union of A and B, denoted A ? B, is the set
  x x ? A ? x ? B
• The intersection of A and B, denoted A ? B, is
  the set x x ? A ? x ? B
• Note If the intersection is void, A and B are
  said to be disjoint.
• The complement of A, denoted , is the set a
  a ? A Note Alternative notation is Ac .If U is
  the universe of discourse then U A

12
Set Operations

• Definitions
• The difference of A and B, or the complement of B
  relative to A, denoted , is the set of
  elements of A that are not elements of B.Note
  The (absolute) complement of A is U - A.
• The symmetric difference of A and B, denoted A ?
  B, is the set (A- B) ? ( B- A).

13
Set Operations

• Example

U 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 A 1, 2,
3, 4, 5, B 4, 5, 6, 7, 8. Then
1, 2, 3, 4, 5, 6, 7, 8 4, 5
0, 6, 7, 8, 9, 10 0, 1, 2, 3, 9, 10 A
- B 1, 2, 3 B - A 6, 7, 8 1,
2, 3, 6, 7, 8
14
Set Operations
15
Set Operations

• Set identities to prove a set identity, several
  methods
• show that each set is a subset of the other one
  (demonstrate in both directions)
• use a set builder to find the correspondance with
  propositional logic and demonstrate using rules
  from propositional logic
• use a membership table similar to a truth table,
  a membership table represents the different
  possibilities for an element to belong to a set
  (1) or not (0).

16
Set Operations

• Set identities

17
Set Operations

• Example of proof
• sets are equal

18
Set Operations

• Example of proof
• set builder

19
Set Operations

• Example of proof
• membership table

20
Set Operations

• Union and intersection of indexed
  collectionsLet A1, A2, , An be an indexed
  collection of sets.Union and intersection are
  associative (because and and or are), we
  have