15.1 Venn Diagrams
Venn Diagram
Venn
Definition of
Diagram
A Venn diagram is a diagram that uses circles to illustrate the relationships
among sets.
Venn diagrams were invented by a guy named John Venn
as a way of picturing relationships between different groups
of things. Since the mathematical term for "a group of
things" is "a set", Venn diagrams can be used to illustrate
both set relationships and logical relationships.
Let's say that our universe contains the
numbers 1, 2, 3, and 4. Let A be the
set containing the numbers 1 and 2;
that is, A = {1, 2}. Let B be the set
containing the numbers 2 and 3; that
is, B = {2, 3}. Then we have the
following relationships, with pinkish
in the Venn diagrams:
set
notation
pronunciation
meaning
AUB
"A union B"
everything
that is in
either of the sets
{1, 2,
3}
"A intersect B"
only the things
that are in
both of the sets
{2}
"A complement",
or "not A"
everything
in the universe
outside of A
{3, 4}
A–B
"A minus B", or
"A complement
B"
everything in A
except for anything
in its overlap with B
{1}
~(A U B)
"not (A union
B)"
everything
outside
A and B
{4}
"not (A intersect
B)"
everything outside
of the overlap
of A and B
{1, 3,
4}
A^B
or
Ac
or
A
~A
~(A ^ B)
or
~(
)
Venn diagram
Shade in A U(B – C)
As usual when faced with parenthesis, I'll work from the inside out.
I'll first find B – C. "B complement C" means I
take B and then throw out its overlap with C,
which gives me this:
Now I have to union this with A:
Shade in ~[(B UC) – A]
when dealing with nested grouping symbols, I'll work from the inside out.
The union of B and C shades both circles fully:
Now I'll do the "complement A" part by cutting out the
overlap with A:
The "not" complement with the tilde
Examples of Venn Diagram
•The Venn diagram below shows the number of grade 5 students
who scored above 50% on Math, above 50% on Science, and above
50% on both Math and Science.
From the Venn diagram, we learn that 16 students scored above
50% on Math, 21 students scored above 50% on Science, and 6
students scored above 50% on both Math and Science.
Solved Example on Venn Diagram
Find the LCM of 14 and 36 using a Venn diagram.
Solution:
Step 1: 14 = 2 x 7 36 = 2 x 2 x 3 x 3
[Write the prime factors of 14 and 36.]
Step 2: Represent all the prime factors of 14 and 36 in a Venn diagram.
Step 3: 7 x 2 x 3 x 3 x 2 = 252 [Multiply all the factors in the Venn diagram.]
Step 4: The LCM of 14 and 36 is 252.
•Out of forty students, 14 are taking English Composition and 29 are
taking Chemistry. If five students are in both classes, how many
students are in neither class? How many are in either class? What is
the probability that a randomly-chosen student from this group is
taking only the Chemistry class?
Suppose I discovered that my cat had a taste for the adorable little
geckoes that live in the bushes and vines in my yard, back when I
lived in Arizona. In one month, suppose he deposited the following
on my carpet: six gray geckoes, twelve geckoes that had dropped
their tails in an effort to escape capture, and fifteen geckoes that
he'd chewed on a little. Only one of the geckoes was gray, chewed
on, and tailless; two were gray and tailless but not chewed on; two
were gray and chewed on but not tailless. If there were a total of
24 geckoes left on my carpet that month, and all of the geckoes
were at least one of "gray", "tailless", and "chewed on", how many
were tailless and chewed on but not gray?
I get that x = 3.
Three geckoes were tailless and chewed on but
not gray
RESTAURANTS
CARTOONS
CONCERT*
STAR TREK
MYTHOLOGY*
POLLUTANTS
TENNIS
STROKES*
TENNIS
TOURNAMENT
MODELS
LANGUAGES
Restaurants
Fill in the Venn Diagram that would represent this data.
100 people seated at different tables in a Mexican restaurant were asked if their
party had ordered any of the following items: margaritas, chili con queso, or
23 people had ordered none of these items.
11 people had ordered all three of these items.
29 people had ordered chili con queso or quesadillas but did not order margaritas.
46 people had ordered at least two of these items.
queso.
26 people had ordered margaritas and chili con queso.
Back
Cartoons
Fill in the Venn Diagram that would represent this data.
A study was made of 200 students to determine what TV shows they
watch.
•22 students don't watch these cartoons.
•73 students watch only Tiny Toons.
•136 students watch Tiny Toons.
•14 students watch only Animaniacs and Pinky & the Brain.
•31 students watch only Tiny Toons and Pinky & the Brain.
•63 students watch Animaniacs.
•113 students do not watch Pinky & the Brain
Back
Concert
Fill in the Venn Diagram that would represent this data.
150 people at a Van Halen concert were asked if they knew how to play piano, drums
or guitar.
18 people could play none of these instruments.
10 people could play all three of these instruments.
77 people could play drums or guitar but could not play piano.
73 people could play guitar.
49 people could play at least two of these instruments.
13 people could play piano and guitar but could not play drums.
21 people could play piano and drums.
Back
Star Trek
Fill in the Venn Diagram that would represent this data.
study was made of 200 students to determine what TV shows they watch.
22 students hated all Star Trek shows.
73 students watch only The Next Generation (TNG).
136 students watch TNG.
14 students watch only Deep Space 9 (DS9) and Voyager (VOY).
31 students watch only TNG and VOY.
63 students watch DS9.
135 students do not watch VOY.
Back
Mythology
Fill in the Venn Diagram that would represent this data.
100 people were asked if they knew who any of the following are: Loki, Hermes, and Ra.
25 people did not know any of these.
3 people knew all three.
48 people knew who Hermes or Ra were but did not know who Loki was.
40 people knew who Ra was.
21 people knew who at least two of these were.
7 people knew who Loki and Ra were but did not know who Hermes was.
8 people knew who Loki and Hermes were.
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Pollutants
Fill in the Venn Diagram that would represent this data.
A study was made of 1000 rivers to determine what pollutants were in them.\\
177 rivers were clean
101 rivers were polluted only with crude oil
439 rivers were polluted with phosphates.
72 rivers were polluted with sulfur compound and crude oil, but not with phosphates.
289 rivers were polluted with phosphates, but not with crude oil.
463 rivers were polluted with sulfur compounds.
137 rivers were polluted with only phosphates.
Back
Tennis Strokes
Fill in the Venn Diagram that would represent this data.
200 tennis players were asked which of these strokes they considered their weakest
stroke(s): the serve, the backhand, the forehand.
20 players said none of these were their weakest stroke.
30 players said all three of these were their weakest stroke.
40 players said their serve and forehand were their weakest strokes.
40 players said that only their serve and backhand were their weakest strokes.
15 players said that their forehand but not their backhand was their weakest stroke.
52 players said that only their backhand was their weakest stroke.
115 players said their serve was their weakest stroke.
Back
Tennis Tournaments
Fill in the Venn Diagram that would represent this data.
200 people were asked which of these grand slam tournaments that they have
attended. The tournaments inquired about were: US Open, Wimbledon, or Australian
Open.
30 people have not attended any of these tournaments.
10 people have been to all thee tournaments.
25 people have been to Wimbledon and the Australian Open.
20 people have been to the US Open and the Australian Open but not to Wimbledon.
65 people have been to exactly two of the tournaments.
165 people have been to the US Open or to Wimbledon.
120 people have been to Wimbledon or to the Australian Open.
Back
Models
Fill in the Venn Diagram that would represent this data.
74 models were hired for the spring shows at a certain Paris design house and the
following information about their body piercing was compiled. Note: Ears count as
one piercing.
Among the models with pierced ears combined with another body piercing, half
the difference between all three parts pierced and only ears and nose pierced
have only their ears and tongue pierced.
There are 10 models who had all three parts pierced.
18 models have multiple piercings.
All but 5 models have pierced ears.
No model has pierced her nose or her tongue without piercing her ears as well,
but one pierced both her nose and her tongue without piercing her ears.
The first statement gives that .5(e-b) = d or this can be written
as e- b = 2d
•The second statement says that e = 10
•The third statement says that e + b + d+ f = 18
•The fourth statement says that h + g + f + c = 5 it also says that
a + b + e + d = 69
•
•The last statement gives the following information.
g=0
f=1
c=0
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Languages
Fill in the Venn Diagram that would represent this data.
There are 208 student at Irion County High School. The foreign language
department offers French, German, and Spanish.
4 students take all three languages.
48 students study French.
There are twice as many students who study both French and Spanish (but not
German) as who study both French and German (but not Spanish), and 4 times as
many as who study all 3.
124 students study Spanish.
27 poor misguided souls do not study any foreign language.
The group of students who study both French and Spanish (but not German) is
exactly the same size as the group made up of students who study both German
and Spanish.
From the first and fifth statement, I have filled in the above numbers
The second statement gives: d + e + f + 4 = 48
The third statement gives two different equations: d = 2 e and d = 4 (4) = 16
The fourth statement gives: a + b + d + 4 = 124
The sixth statement gives: d = 4 + b
Remember that the total number of students is 208
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ASSIGNMENT
• Page 568 4,8,12,14, 19-22