Download document 6974986

Section 2-1 & 2-2
Symbols and Terminology
Venn Diagrams and Subsets
Lesson Objectives
• Distinguish between finite and infinite sets.
• Describe elements of number sets.
• Use Venn diagrams to depict set
relationships.
• Determine the complement of a set within a
universal set.
• Determine if one set is a subset of another.
• Understand the distinction between a subset
and a proper subset.
Do Now
1. List the names of your siblings.
2. Create two more lists, one with the names of
your male siblings and the other with the names
of your female siblings.
Definition
A set is a collection of objects. The objects
belonging to the set are called the elements, or
members, of the set.
Questions to Consider
1. Does order matter?
2. Can an element show up more than once?
3. What if a set contains nothing?
Definition
The set containing no elements is called the
empty set (null set) and is denoted by { } or .
Sets of Numbers
Natural numbers (counting) {1, 2, 3, 4, …}
Whole numbers {0, 1, 2, 3, 4, …}
Integers {…,–3, –2, –1, 0, 1, 2, 3, …}
p

Rational numbers  p and q are integers, with q  0
q

May be written as a terminating decimal, like 0.25,
or a repeating decimal, like 0.333…
Irrational {x | x is not expressible as a quotient of integers}
Decimal representations never terminate and never
repeat.
Real numbers {x | x can be expressed as a decimal}
Example: Describing Sets
Word description:
The set of positive even integers less than 10
The listing method:
{2, 4, 6, 8}
Set-builder notation:
{x|x is a positive even integer less than 10}
Example: List the sets
1. The set of all positive multiples of 3.
2. The set of counting numbers between 4 and 14.
Definition
The number of elements in a set is called the
cardinal number, or cardinality, of the set.
The symbol n(A), read “n of A,” represents the
cardinal number of set A.
Example: Finding Cardinal Numbers
Find the cardinal number of each set.
a) K = {red, green, blue, yellow}
b) M = {2}
c) 
d) L={ x| x is an even integer}
Definition
In set theory, the universal set is the set of all
elements under consideration, typically designated
with the letter U.
Venn Diagrams
The rectangle represents the universal set, U,
while the portion bounded by the circle
represents set A.
A
U
Complement of a Set
The shaded region inside U and outside the circle
is labeled A' (read “A prime”). This set, called the
complement of A, contains all elements that are
contained in U, but not in A.
A
A
U
Subsets of a Set
Set A is a subset of set B if every element of A is
also an element of B. In symbols this is written A  B.
B
A
U
Proper Subset of a Set
Set A is a proper subset of set B if A  B
and A  B.
In symbols, this is written A  B.
Example: Determining Subsets and Proper
Subsets
Consider A={a,c,e}
Which of the following is a subset of A? proper
subset of A?
D={a}
E={b,f}
F={a,c,e}
Can you describe the set of ALL subsets of A?
Counting Subsets
One method of counting subsets involves using
a tree diagram. The figure below shows the use
of a tree diagram to find the subsets of {a, b}.
a subset? b subset? 4 subsets
Yes
No
Yes
No
Yes
No
{a, b}
{a}
{b}

Number of Subsets
The number of subsets of a set with n elements
is 2n.
The number of proper subsets of a set with
n elements is 2n – 1.
Example: Finding Numbers of Subsets and
Proper Subsets
Find the number of subsets and the number of
proper subsets of the set {m, a, t, h, y}.
Solution
Since there are 5 elements, the number of
subsets is 25 = 32.
The number of proper subsets is 32 – 1 = 31.