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Math
A |
Venn
Diagrams
(Sets) |
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A Venn
diagram is a drawing, in which circular areas represent groups of
items sharing common properties. The drawing consists of two or more
circles, each representing a specific group. 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 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. |
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The
Venn diagram at the left shows two sets
A
and
B.
Values that belong to both set
A
and set
B
are located in the center region labeled
A ^ B
where the circles overlap.
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The notation
A v B
represents the entire region covered by both sets
A
and
B.
If we cut out sets
A
and
B,
the remaining region in
U
would be labeled
~(A v B). |
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The most interesting
features of Venn diagrams are 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
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x
= 3
(There are 3 dogs which are black with long hair but do not have a
short tail.)
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