Set Concepts

• A set is a collection of objcts.
• The objects in a set are called elements or members
• A = { 1, frog, square, x}
• A set is well defined if the contents can be clearly determined.
• If a collection of objects is not well definied, our authors say it is not a set. It is a collection.
  • The collection of the last three presidents of the US.
  • The collection five best top 40 hits.
• A set is finite if the number of elements in the set is a natural number.
  • Otherwise the set is infinite
  • Infinite sets are well defined: The integers is a well defined set.
  • If you give an example of a finite set, use something that is clear:
  • The number of years the universe has existed, or will exist is unknown.
  • The number of stars in the sky.
• Set notation:
  • Roster Form, just list the elements.
    • You need {}.
    • A = { 1, 2, 3, 4}
    • We use dots to show that we follow the pattern
    • B = { 1, 2, 3, ..., 10}
    • Or for infinate sets
    • N = { 1, 2, 3, ...}, these are the natural numbers.
  • Descriptive, Describe the set.
    • The natural numbers between 5 and 8 = { 6, 7}
    • The natural numbers between 5 and 8 inclusive = {5, 6, 7, 8}
  • Set builder notation
    • X = { x | condition}
    • A = {x | x ∈; N and x < 5}
    • B = {x | x∈; N and 15 ≤x< 119}
    • C = {x | x is an even natural number.}
• x ∈; A is read x is an element of A and means that x is in the set A.
  • A = {1, 2, 3, 4, 5}
  • Is 4 ∈; A?
  • Is 7 ∈; A?
  • Is A ∈; A?
  • Is {4} ∈; A?
• The cardinal number of a set A, written n(A), is the number of elements in set A.
• A set A is equal to a set B, written A=B if and only if the sets A and B contain exactly the same element.
• A set A is equivalent to a set B if n(A) = n(B)
• The empty set is written {}, or ∅, and is a set which contains no element.
• The universal set, written U, is the set of all elements under consideration.

Subsets

• A set A is a subset of a set B if every element of a is an element of B. (A ⊆ B)
• A set A is a proper subset of a set B (A ⊂ B) if A ⊆ B but n(A) &neq; n(B). (B contains more elements than A).
• The empty set is a subet of all sets.
• The set A, n(A) = t, will have 2^t subsets.
• Do a pizza problem.