Transcript MTH 231

MTH 231
Section 2.1
Sets and Operations on Sets
Overview
• The notion of a set (a collection of objects) is
introduced in this chapter as the primary way
to describe whole numbers.
• Operations (e.g., union and intersection) on
sets form the basis for addition, subtraction,
multiplication, and division.
• We will explore different models of these
operations, and subsequently properties of
whole numbers through these models.
Sets
• A set is a collection of objects.
• An object that belongs to a particular set is
called an element, or member.
• Sets must be well-defined:
1. there must be a universe of objects that are
allowed into consideration;
2. each object either is or is not an element of
the set.
Three Ways To Define A Set
1. Word Description.
The letters in the word “alabama”
2. Listing in Braces.
{a, l, b, m}
3. Set-builder Notation.
{x | x is one of the letters in the word
“alabama”}
More
• The order in which elements are listed is
arbitrary.
• Elements should be listed just once.
• Capital letters are generally used to denote, or
name, sets.
• Membership in a set is represented by ϵ.
Venn Diagrams
• A pictorial representation of sets.
• The universal set, denoted by U, is
represented by a rectangle.
• Any sets under discussion are represented by
loops inside the rectangle.
• The region inside the loop is associated with
the elements of the set.
Pictures
Complement
• The complement of a set A is all the elements
in the universal set U that are not elements in
set A.
Subset
• The set A is a subset of another set B if, and
only if every element of A is also an element
of B.
A B
• A is a proper subset of B if A is a subset of B
but A and B are not equal (two sets are equal
if they have precisely the same elements).
Empty Set
• The empty set is a set with no elements.


Intersection
• The intersection of two sets A and B is the set
of elements common to both A and B.
A B
Disjoint Sets
• Two sets A and B are disjoint if A and B have
no elements in common (or, that the
intersection of A and B is the empty set).
A B  
Union
• The union of two sets A and B is the set of
elements that are in A or B (or both).
A B
An Example
• p. 75 #9