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Venn Diagrams. EQ: How do I use a Venn diagram to represent different sets of numbers and to solve problems?. B. A. U. A. B. U. A. B. U. Disjoint Sets. U. A B =. Disjoint and Exhaustive. U = {x|-3<x<8, x } A and B are subsets of U, A={x|x<5} and B = {x|x 5}.

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venn diagrams

Venn Diagrams

EQ: How do I use a Venn diagram to represent different sets of numbers and to solve problems?

slide2
B

A

U

slide3
A

B

U

slide4
A

B

U

disjoint and exhaustive
Disjoint and Exhaustive

U = {x|-3

A and B are subsets of U, A={x|x<5} and B = {x|x 5}

four regions on a venn diagram
A

B

U

Four regions on a Venn Diagram
  • In A, but not in B
  • In B, but not in A
  • In both A and B
  • Neither in A nor B
example 1
Q

P

(7)

(11)

(3)

(4)

U

Example 1
  • Given the Venn Diagram below, how many elements are there in
  • P
  • P U Q
  • Q’
  • P, but not Q
  • Q, but not P
  • neither P nor Q?
you do
Y

X

(8)

(3)

(6)

(2)

U

YOU DO:
  • Give the number of elements in:
  • X’
  • X Y
  • X U Y
  • X, but not Y
  • Y, but not X
  • Neither X nor Y
example 2
B

A

6 + c = 17 ; c = 11

11

Example 2:

8 + 6 + 11 + d = 30

25 + d = 30 ; d = 5

5

  • Given n(U) = 30, n(A) = 14,

n(B) = 17 and n(A B) = 6 find:

  • n(A U B)
  • n(A, but not B)

25

8

(a)

(c)

  • b = 6
  • a + b = 14
  • b + c = 17
  • a + b + c + d = 30

6

(b)

a + 6 = 14

a = 8

8

(d)

U

you do1
B

A

(a)

(c)

(b)

(d)

U

YOU DO:
  • Given n(U) = 26, n(A) = 11, n(B) = 12 and n(A B) = 8, find:
  • n(A U B)
  • n(B, but not A)
  • n(A’)

15

4

15

b = 8

a + b = 11; a = 3

b + c = 12; c = 4

a + b + c + d = 26

3 + 8 + 4 + d = 26; d = 11

now the real thing
Black

Brown

(a)

(c)

(b)

(d)

U

Now, the real thing…
  • A squash club has 27 members. 19 have black hair, 14 have brown eyes and 11 have both black hair and brown eyes.
    • Place this information on a Venn Diagram
    • Find the number of members with:
      • Black hair or brown eyes
      • Black hair, but not brown eyes

d = 5

a + b + c + d = 27

a + b = 19

b + c = 14

a = 8

c = 3

b = 11

you do2
Long

Brown

(a)

(c)

(b)

(d)

U

YOU DO:

Pele has 14 cavies as pets. Five have long hair and 8 are brown. Two are both brown and have long hair.

  • Place this information on a Venn diagram
  • Find the number of cavies that:
    • Are short haired
    • Have short hair and are brown
    • Have short hair and are not brown

c + d = 9

c = 6

d = 3

a + b + c + d = 14

a + b = 5

b + c = 8

b = 2

d = 3

a = 3

c = 6

a little bit different
10 m

4 m

(a)

(c)

(b)

(d)

U

A little bit different…
  • A platform diving squad of 25 has 18 members who dive from 10 m and 17 who dive from 4 m. How many dive from both platforms?
  • a + b + c + d = 25
  • a + b = 18
  • b + c = 17
  • 18 + c + 0 = 25
  • c = 7
  • Therefore b + 7 = 17 and b = 10
now for the real real thing
Now for the real real thing…
  • A city has three newspapers A, B, and C. Of the adult population, 1% read none of these newspapers, 36% read A, 40% read B, 52% read C, 8% read A and B, 11% read B and C, 13% read A and C and 3% read all three papers. What percentage of the adult population read:
  • Newspaper A only
  • Newspaper B or Newspaper C
  • Newspaper A or B but not C

A

B

d

d

g

g

e

a = 3; a + d = 8; a + b = 11; a + c = 13

Therefore, d = 5, b = 8, and c = 10

g + 3 + 5 + 10 = 36; therefore g = 36

e + 3 + 5 + 8 = 40; therefore e = 24

f + 3 + 8 + 10 = 32; f = 31 and h = 1

a

a

c

c

b

b

b

b

b

f

f

f

C

U

U

U

slide18
HW #4
  • 4a – pg 78 #1; #4; pg 79 #7,

pg 80 #9

  • 4b – pg 81, 82 #11 - 14
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