Download Sets and Venn Diagrams

Definition: A set is a well-defined
collection of objects.
Sets and Venn
Diagrams
Typically, we will refer to sets of
numbers.
Notation: Mathematically, sets are
collected by squiggly brackets.
Example: {7, 24, 2.5} is a set.
More notation:
Definition: What a set contains is called
its members or elements.
Example: 24 is an element (or a member)
of {7, 24, 2.5} .
∈, which looks like a stylized E, stands
for the words “is an element of”.
Example: 24 ∈ {7, 24, 2.5} .
∉ stands for the words “is not an
element of”.
Example: 2 ∉ {7, 24, 2.5} .
More notation: Just like variable names
can stand for numbers, we can assign
names to sets.
Typically, sets are assigned a capital
letter for their name.
Example: A = {7, 24, 2.5}
24 ∈ A
2∉A
Definition: A set that contains no elements
is called the null set or empty set.
{ } is the null or empty set.
Notation: ∅ represents the empty set.
∅={}
It is important to note that the empty set
contains nothing. A set that contains zero
contains something!
∅≠{0}
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Notation: For A a set, n(A) means “the
number of elements in the set A”.
Examples: If A = {7, 24, 2.5}, then
n(A) = 3.
For B = { 0 }, n(B) = 1, while n(∅) = 0.
In fact, for any set, C, if n(C) = 0, then
C = ∅.
Some sets are too big to list every
element. To describe those sets, we
could list enough elements to develop a
pattern, followed by an ellipsis (dot, dot,
dot):
Sets are equal if they contain the same
elements, not matter what order.
Example:
{7, 24, 2.5} = {24, 7, 2.5} = {2.5 7, 24},
etc.
Another way to describe a set is with set
builder notation, and the symbol |, which is
read “such that”.
{x | x is a positive, odd number} =
{1, 3, 5, 7, . . .}
Example: {1, 3, 5, 7, . . .} is the set of
all positive, odd numbers.
In English, we would read {x | x is a
positive, odd number} as “the set of all x
such that x is a positive, odd number”.
The universal set is a set that contains
every one of the kind of objects in a set.
The universal set is not necessary unique,
and should be defined in each problem.
Definition: If A and B are sets such that
all the elements of A are also in B, then we
say that A is a subset of B.
Example: If A = {7, 24, 2.5}, then the
universal set could be all real numbers, or
it could be all rational numbers.
Example: If A = {7, 24, 2.5} and
B = { 24, 2.5, 15, -3, 7 }, then A ⊆ B .
Notation: A ⊆ B.
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Definition: If A and B are sets and A is a
proper subset of B, then A is a subset of B,
but A is not equal to B.
That is, A ⊆ B , A ≠ B.
Notation: A ⊂ B (A is a proper subset of
B).
All proper subsets are subsets, but not all
subsets are proper.
Note: The empty set is a subset of every
set.
A counting problem:
How many subsets are there for {1, 2}?
We can list these subsets:
{1}, {2}, {1, 2}, and ∅, so {1, 2} has 4
subsets.
Tree-diagram of all possible subsets of {1, 2}
contains 1?
yes
contains 2?
subset is:
yes
{1, 2}
no
{1}
Example: If A = {7, 24, 2.5} ,
B = { 24, 2.5, 15, -3, 7 }, and
C = {2.5, 24, 7}, which of the following
is (are) true?
1. A ⊆ B
5. B ⊂ A
2. A ⊂ B
6. B = C
3. A ⊆ C
7. A = C
4. A ⊂ C
How many subsets of a set containing 3
elements? How about 8 elements? To
develop a formula for the numbers of
subsets of a set containing any number of
elements, we will use a tree diagram.
A tree diagram is a method of keeping
track of all possible outcomes:
Note that for every element in the set, you
must answer the same 2 questions, and
each answer represents another possible
subset.
That is, for each element in the set, we
double the number of possible subsets.
no
yes
{2}
no
∅
So, a set with n distinct elements has 2n
possible subsets.
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A Venn diagram is a graphical
representation of the relation among sets.
Example: How many subsets does the set
{1, 3, 5, 7} have?
Example: Let A = {1, 2}, B = {1, 2, 5, 7}.
Note that both A and B contain integers
only, so let U = {x | x is an integer} be the
universal set. The following Venn diagram
represents those sets:
A
Example: Use the Venn diagram to state
the relationship between C and D.
B
A⊆B⊆U
U
Definition: For any set A, with universal
set U, then every element of U that is not in
A is in the complement of A.
Notation: A’ = {x | x ∈ U and x ∉ A}
D
C
U
Venn diagram:
A’
A
Definition: For sets A and B, the
intersection of A and B is the sets of all
points in both A and B.
Definition: A and B are disjoint sets if
A ∩ B = ∅.
Notation: A ∩ B = { x | x ∈ A and x ∈ B}
A
B
A
B
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Definition: For sets A and B, the union of
A and B is the set of all points either in set
A or in set B.
We’ll read A ∩ B as either “A intersection
B” or as “both A and B”.
Notation: A ∪ B = { x | x ∈ A or x ∈ B}
A
B
We’ll read A ∪ B as either “A union B” or
as “either A or B”.
Classwork: For A = { 3, 15, 27, 9},
B = { 1, 2, 9, 3, 4, 15, 19, 27}, and
C = { 2}
1. Find A ∩ B, or state that the sets are
disjoint.
2. Find A ∩ C, or state that the sets are
disjoint.
3. Find A ∪ C.
4. If B is the universal set, U, find C ‘.
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