DtoI05 Discrete Math Rosen Section 1'67

1
DtoI05 Discrete MathRosenSection 1.6-7

• Alan Coppola
• Richard Weiss
• http//grace.evergreen.edu/dtoi

2
Sec 1.6

• Set collection of elements
• Subset a sub collection of elements
• Empty set a subset of each set
• Proper Subset A ? B, but A ? B, denoted by A ?
  B
• Finite Set of Cardinality n A set with exactly
  n distinct elements.
• The size or cardinality of a set S is denoted by
  S. If it can be put into one-to-one
  correspondence with the first n counting numbers,
  it is said to be of size n.

3
Sec 1.6 - continued

• A set S is said to be infinite if it is not
  finite.
• Question Can you prove that there is an infinite
  set ?

4
Sec 1.6 - Contd

• An ordered n-tuple (a1, a2, a3, , an ) is the
  ordered collection or list that has a1 has as
  its first element, a2 as its second element, ,
  and an as its nth element.
• Ordered pairs (a1, a2 ) and (b1, b2 ) are equal
  if and only if a1 b1 and a2 b2
• Let A and B be sets. The Cartesian product of A
  and B is denoted by AxB, is the set of all
  ordered pairs (a, b), where a ? A and b ? B.
• A x B (a,b) a? A ? b? B
• A1 x A2 x x An (a1, a2, a3, , an ) ai ?
  Ai , for I 1, 2, , n

5
Sec 1.7 Set Operations

• ????????????
• Union - A? B x x ? A ? x ? B
• Intersection A? B x x ? A ? x ? B
• Disjoint A? B ?
• Venn Diagram
• Inclusion-Exclusion A? B A B - A?
  B
• Difference A B x x ? A ? x ? B

6
Sec 1.7 Set Operations

• Universe universal setdomain of discourse
• A- complement of A x x ? A

7
Sec 1.7 Set Identities

• Identity A ? ? A, A ? U A
• DominationA ? U U, A ? ? ?
• Idempotent A ? A A, A ? A A
• Complementation A A
• Commutative A ? F F ? A, A ? F F ? A
• Associative A ? (B ? F) (A ? B) ? F A ? B ?
  F
• Distributive (A ? (B ? F)) (A ? B) ?(A ? F)
  (A ? (B ? F)) (A ? B) ? (A ? F)
• De Morgans (A ? F) A ? F, (A ? F)
  A ? F
• Absorption (A ? (A ? F)) A, (A ? (A ? F))
  A
• Complement Law A ? A U, A ? A ?

8
Sec 1.7 Identity Proof

• Problem 9Let A and F be sets. Show that (A ?
  (A ? F)) A