| Set Theory |
| Sets |
A set is a list
or collection
of unique objects
in no particular order.
The
individual objects
of the set are
called elements.
E.g. the set of all prime numbers.
Sets
can be defined in words, or by listing the elements
between curly
braces separated
by commas, or between curly braces containing some
other defining symbols.
For
example, P is a set
of odd numbers between 1 and 20 can be written like
this,
P
= {1, 3, 5, 7, 9, 11, 13, 15, 17, 19}
This
set contains
ten members or its cardinality is 10. The name of
the set is P.
Universal Set
The Universal set is the set of all elements under consideration.
That
is,
U
= {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
Empty Set
A set with no elements is called
an empty set or
a null set and
is denoted by many symbols like {},  ,
or
Finite
and Infinite Sets
Finite
Sets: A finite set is a set where there
are no elements or where the number of elements are countable. For example, the set of even numbers
between 1 and 10 is a finite set.
Infinite
Sets: An infinite set is
a set where the
number of elements in the set cannot
be counted. For example, the set of all odd numbers.
Equivalent
Sets
Two finite sets are equivalent if they contain the same number of elements.
For example,
A
= {a, b, c, d} B = {f, g, h, i}
Both
A and B contain 4 elements and so are considered equivalent.
Equal
Sets
Two sets are equal if they contain exactly the same elements.
For example,
T
= {3, 6, 9, 12} M = {3, 6, 9, 12}
Both
T and M contain the elements 3, 6, 9, 12 and so are considered equal.
|
| Venn
Diagrams |
|
A Venn diagram is a way of representing sets visually.
In
the diagram to the left we are shown two sets,
A and B, which depicts the section which wholly belongs
to A, the section which wholly belongs to B, the section
which is common to both A and B and the section which
does not belong to both A and B. |
Union
|
The
Venn Diagram Representation of Union |
|
The union of two sets A and B
is the set obtained
by combining the members of each without repetition.
The symbol "U" is used to represent union.
For
example,
A
= {a, b, c} B = {c, d, e}
A U B = (a, b, c, d, e) |
Intersection
|
The
Venn Diagram Representation of Intersection |
|
The intersection of two sets A
and B is the set of elements common
to both. The symbol " "
is used to represent intersection .
For
example,
A
= {a, b, c} B = {c, d, e}
A B = {c} |
Subset
 |
Venn
Diagram Representation of Subset |
|
A subset is a set whose members are members of
another set or
it can be described as a set contained within another set.
The symbol " "
is used to represent subset.
For
example, set B is contained in A since 4, 8 and 12
are common to those elements 2, 4, 6, 8, 10, and 12 which exist in set A
U
= {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}
A
= {2, 4, 6, 8, 10, 12} B = {4, 8, 12}
B A = {4, 8, 12} |
Complement
|
Venn
Diagram Representation of Complements |
|
The complement of
any set is simply
said to be the elements that are not members of a specific set.
In
the diagram to the left, the universal set is
U
= {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
A=
{1, 3, 5, 7, 9,}B ={6, 7, 8, 9, 10}
A
U B = {1, 3, 5, 6, 7, 8, 9, 10}
A B = {7, 9}
A'
= {2, 4, 6, 8, 10}
B'
= {1, 2, 3, 4, 5}
(A
U B)' = {2, 4} |
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