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Sets
are simply collections of items.
A set may contain your favorite even numbers, the days of the week, or
the names of your brothers and sisters. The items contained within
a set are called elements, and
elements in a set do not
"repeat".
The elements of a set are often listed by
roster.
A roster
is a list of the elements in a set,
separated by commas and surrounded by French curly
braces.
For information on other methods of describing sets, see
Set-Builder and Interval Notation.
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Let B = {1, 2, 3,
4, 5, 6, 7, 8, 9, 10}
A = {3, 6, 9}
Set A is a
subset
of set
B, since every element in set A is also an
element of set B. The notation is:
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The empty
set is denoted with the symbol:
Sets are often
represented in pictorial form with a circle containing the elements of the set.
Such a
depiction is called a Venn Diagram.
A Venn
diagram is a drawing, in which circular areas represent groups of
items usually sharing common properties. The drawing consists of two or more
circles, each representing a specific group or set. This process of visualizing logical
relationships was devised by John Venn (1834-1923).
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Each
Venn diagram begins with a
rectangle representing the universal
set. Then each
set of values in the problem is represented by a circle. Any
values that belong to more than one set will be placed in
the sections where the circles overlap.
The
universal set is often the "type" of values that are
solutions to the problem. For example, the universal
set could be the set of all integers from -10 to +10, set A
the set of positive integers in that universe, set B the set
of integers divisible by 5 in that universe, and set C the
set of elements -1, - 5, and 6. |
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The
Venn diagram at the left shows two sets
A
and
B
that overlap.
The universal set is
U. Values that belong to both set
A
and set
B
are located in the center region labeled
where the circles overlap. This region is called
the "intersection" of the two
sets.
(Intersection, is only where the two sets intersect, or
overlap.)
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The notation
represents the entire region covered by both sets
A
and
B
(and the section where they overlap). This region is
called the "union" of the
two sets.
(Union, like
marriage, brings all of both sets together.)
If we cut out sets
A
and
B
from the picture above,
the remaining region in
U,
the universal set,
is labeled ,
and is called the complement
of the union of sets
A
and
B.
A complement of a set is all of the elements (in the
universe) that are NOT in the set.
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NOTE*:
The
complement
of a set can be represented with several
differing notations.
The complement of set A can be written as
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* A
statement from the NY SED says that students should be
familiar with all notations for complement of a set.
The
SED
Glossary shows the first two notations, while the
SED
Sample Tasks show the third.
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Example:
Let U (the
universal set) = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
(a
subset of the positive integers)
A = {2, 4, 6, 8}
B = {1, 2, 3, 4, 5}
Union - ALL elements in BOTH sets
Intersection - elements where sets overlap
Complement - elements NOT in the set
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**
One of the most interesting
features of Venn diagrams is the areas or sections where
the circles overlap one another -- implying that a
sharing is occurring. |
This ability to represent a "sharing of conditions"
makes Venn diagrams
useful tools for solving complicated problems. Consider the
following example:
Example:
| Twenty-four
dogs are in a kennel. Twelve of the dogs are black,
six of the dogs have short tails, and fifteen of the dogs
have long hair. There is only one dog that is black
with a short tail and long hair. Two of the dogs are
black with short tails and do not have long hair.
Two of the dogs have short tails and long hair but are not
black. If all of the dogs in the kennel have at
least one of the mentioned characteristics, how many dogs are black with long hair but do
not have short tails? |
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Solution:
Draw a
Venn diagram to represent the situation described in the problem.
Represent the number of
dogs that you are looking for with
x.
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Notice
that the number of dogs in each of the three categories is labeled
OUTSIDE of the circle in a colored box. This number is a
reminder of the total of the
numbers which may appear anywhere inside that particular circle.
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After
you have labeled all of the conditions listed in the problem, use this
OUTSIDE box number to help you determine how many dogs are to be labeled
in the empty sections of each circle.
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Once
you have EVERY section in the diagram labeled with a number or an
expression, you are ready to solve the problem.
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Add
together EVERY section in the diagram and set it equal to the total
number of dogs in the kennel (24). Do NOT use the OUTSIDE
box numbers.
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9
- x + 2 + 1 + 1 + 2 + x + 12 - x = 24
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27
- x = 24
-
x
= 3
(There are 3 dogs which are black with long hair but do not have a
short tail.)
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(5 is
not in set A)
