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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?


B

A

U


A

B

U


A

B

U



Disjoint and exhaustive
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}


U 1 2 3 4 5 6 7 8 a 1 3 6 8 b 2 3 4 5 8
U={1,2,3,4,5,6,7,8}A = 1,3,6,8 B = {2,3,4,5,8}

A

B

U


U 1 2 3 4 5 6 7 8 a 1 3 6 7 8 b 3 6 8
U={1,2,3,4,5,6,7,8}A = {1,3,6,7,8} B = {3,6,8}

A

B

U


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


HW #4

  • 4a – pg 78 #1; #4; pg 79 #7,

    pg 80 #9

  • 4b – pg 81, 82 #11 - 14


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