Venn Diagrams and Subsets
MATH 100 Survey of Mathematical Ideas
J. Robert Buchanan
Department of Mathematics
Fall 2014
J. Robert Buchanan
Venn Diagrams and Subsets
Venn Diagrams
When discussing sets there is a stated or implied universe
of discourse which includes all things under study in a
given context.
The universe of discourse is called the universal set and
is denoted U.
The universal set may change depending on the topic of
discussion.
A Venn diagram represents the universal set as a
rectangle and other sets within the universal set as circles.
J. Robert Buchanan
Venn Diagrams and Subsets
Example
U
A
J. Robert Buchanan
Venn Diagrams and Subsets
Complement of a Set
Definition
For any set A within the universal set U, the complement of A
denoted A0 is the set of elements in U that are not elements of
A.
A0 = {x | x ∈ U and x ∈
/ A}
J. Robert Buchanan
Venn Diagrams and Subsets
Complement of a Set
Definition
For any set A within the universal set U, the complement of A
denoted A0 is the set of elements in U that are not elements of
A.
A0 = {x | x ∈ U and x ∈
/ A}
Let U = {a, b, c, d, e, f , g, h} and C = {b, e}, then the
complement of C is
C 0 = {a, c, d, f , g, h}.
J. Robert Buchanan
Venn Diagrams and Subsets
Example of Set and Complement
U
A'
A
J. Robert Buchanan
Venn Diagrams and Subsets
Examples
Let U = {a, b, c, d, e, f , g} and A = {a, e} and B = {b, e, f }.
Find the following sets and enter them as alphanumeric
responses. Do not enter the “curly braces” ({ }) and separate
set elements with spaces instead of commas. For the empty
set enter NULL SET.
A0
J. Robert Buchanan
Venn Diagrams and Subsets
Examples
Let U = {a, b, c, d, e, f , g} and A = {a, e} and B = {b, e, f }.
Find the following sets and enter them as alphanumeric
responses. Do not enter the “curly braces” ({ }) and separate
set elements with spaces instead of commas. For the empty
set enter NULL SET.
A0
B0
J. Robert Buchanan
Venn Diagrams and Subsets
Examples
Let U = {a, b, c, d, e, f , g} and A = {a, e} and B = {b, e, f }.
Find the following sets and enter them as alphanumeric
responses. Do not enter the “curly braces” ({ }) and separate
set elements with spaces instead of commas. For the empty
set enter NULL SET.
A0
B0
U0
J. Robert Buchanan
Venn Diagrams and Subsets
Subsets
Definition
A set A is a subset of set B if every element of A is also an
element of B. This relationship is denoted A ⊆ B.
J. Robert Buchanan
Venn Diagrams and Subsets
Subsets
Definition
A set A is a subset of set B if every element of A is also an
element of B. This relationship is denoted A ⊆ B.
To express that set A is not a subset of set B we may write
A 6⊆ B.
J. Robert Buchanan
Venn Diagrams and Subsets
Examples
Let U = {a, b, c, d, e, f , g} and A = {a, e} and B = {b, e, f }.
Determine whether the following statements are true or false.
Select button “A” for true and button “B” for false.
A⊆U
J. Robert Buchanan
Venn Diagrams and Subsets
Examples
Let U = {a, b, c, d, e, f , g} and A = {a, e} and B = {b, e, f }.
Determine whether the following statements are true or false.
Select button “A” for true and button “B” for false.
A⊆U
A⊆B
J. Robert Buchanan
Venn Diagrams and Subsets
Examples
Let U = {a, b, c, d, e, f , g} and A = {a, e} and B = {b, e, f }.
Determine whether the following statements are true or false.
Select button “A” for true and button “B” for false.
A⊆U
A⊆B
B⊆U
J. Robert Buchanan
Venn Diagrams and Subsets
Examples
Let U = {a, b, c, d, e, f , g} and A = {a, e} and B = {b, e, f }.
Determine whether the following statements are true or false.
Select button “A” for true and button “B” for false.
A⊆U
A⊆B
B⊆U
∅⊆B
J. Robert Buchanan
Venn Diagrams and Subsets
Set Equality and Proper Subsets
Remark: every set is a subset of itself, i.e., A ⊆ A, thus we may
present an alternative definition of set equality.
J. Robert Buchanan
Venn Diagrams and Subsets
Set Equality and Proper Subsets
Remark: every set is a subset of itself, i.e., A ⊆ A, thus we may
present an alternative definition of set equality.
Definition
Suppose A and B are sets. If A ⊆ B and B ⊆ A, then A = B.
J. Robert Buchanan
Venn Diagrams and Subsets
Set Equality and Proper Subsets
Remark: every set is a subset of itself, i.e., A ⊆ A, thus we may
present an alternative definition of set equality.
Definition
Suppose A and B are sets. If A ⊆ B and B ⊆ A, then A = B.
Definition
Set A is a proper subset of set B if A ⊆ B and A 6= B.
Symbolically we represent this relationship as A ⊂ B.
J. Robert Buchanan
Venn Diagrams and Subsets
Example
Determine whether S ⊂ T , S ⊆ T , both, or neither is
appropriate for the following sets. On your i>clicker press
A: for S ⊂ T
B: for S ⊆ T
C: for both S ⊂ T and S ⊆ T
D: for neither S ⊂ T nor S ⊆ T
S
{M, W , F }
J. Robert Buchanan
T
{S, M, T , W , Th}
Venn Diagrams and Subsets
Example
Determine whether S ⊂ T , S ⊆ T , both, or neither is
appropriate for the following sets. On your i>clicker press
A: for S ⊂ T
B: for S ⊆ T
C: for both S ⊂ T and S ⊆ T
D: for neither S ⊂ T nor S ⊆ T
S
{M, W , F }
{a, n, d}
J. Robert Buchanan
T
{S, M, T , W , Th}
{r , a, n, d, y }
Venn Diagrams and Subsets
Example
Determine whether S ⊂ T , S ⊆ T , both, or neither is
appropriate for the following sets. On your i>clicker press
A: for S ⊂ T
B: for S ⊆ T
C: for both S ⊂ T and S ⊆ T
D: for neither S ⊂ T nor S ⊆ T
S
{M, W , F }
{a, n, d}
∅
J. Robert Buchanan
T
{S, M, T , W , Th}
{r , a, n, d, y }
∅
Venn Diagrams and Subsets
Example
Determine whether S ⊂ T , S ⊆ T , both, or neither is
appropriate for the following sets. On your i>clicker press
A: for S ⊂ T
B: for S ⊆ T
C: for both S ⊂ T and S ⊆ T
D: for neither S ⊂ T nor S ⊆ T
S
{M, W , F }
{a, n, d}
∅
{2, 1/3, 5/9}
J. Robert Buchanan
T
{S, M, T , W , Th}
{r , a, n, d, y }
∅
Q
Venn Diagrams and Subsets
How Many Subsets Does a Set Have?
List all the subsets of the following sets:
A = {a, b, c}
J. Robert Buchanan
Venn Diagrams and Subsets
How Many Subsets Does a Set Have?
List all the subsets of the following sets:
A = {a, b, c}
Subsets: ∅, A, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}
J. Robert Buchanan
Venn Diagrams and Subsets
How Many Subsets Does a Set Have?
List all the subsets of the following sets:
A = {a, b, c}
Subsets: ∅, A, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}
B = {2, 4, 6, 8}
J. Robert Buchanan
Venn Diagrams and Subsets
How Many Subsets Does a Set Have?
List all the subsets of the following sets:
A = {a, b, c}
Subsets: ∅, A, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}
B = {2, 4, 6, 8}
Subsets: ∅, B, {2}, {4}, {6}, {8}, {2, 4}, {2, 6}, {2, 8},
{4, 6}, {4, 8}, {6, 8}, {2, 4, 6}, {2, 4, 8}, {2, 6, 8}, {4, 6, 8}
J. Robert Buchanan
Venn Diagrams and Subsets
How Many Subsets Does a Set Have?
List all the subsets of the following sets:
A = {a, b, c}
Subsets: ∅, A, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}
B = {2, 4, 6, 8}
Subsets: ∅, B, {2}, {4}, {6}, {8}, {2, 4}, {2, 6}, {2, 8},
{4, 6}, {4, 8}, {6, 8}, {2, 4, 6}, {2, 4, 8}, {2, 6, 8}, {4, 6, 8}
Question: can you conjecture (using inductive reasoning) the
number of subsets of a given set?
J. Robert Buchanan
Venn Diagrams and Subsets
Numbers of Subsets and Proper Subsets
Remark:
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.
J. Robert Buchanan
Venn Diagrams and Subsets