Venn DiagramsVenn Diagrams

Numbers in each regionNumbers in each region

Example 1 Example 1

Given the Venn Given the Venn Diagram below, Diagram below, how many how many elements are there elements are there in in

a)a) PP

b)b) P U Q P U Q

c)c) Q’Q’

d)d) P, but not QP, but not Q

e)e) Q, but not PQ, but not P

f)f) neither P nor Q? neither P nor Q?

P

(3)(11)

(4)

(7)

Q

U

YOU DO: YOU DO:

Give the number Give the number of elements in: of elements in:

a)a) X’X’

b)b) X YX Y

c)c) X U Y X U Y

d)d) X, but not YX, but not Y

e)e) Y, but not XY, but not X

f)f) Neither X nor YNeither X nor Y

X

(6)(3)

(2)

(8)

Y

U

Example 2: Example 2:

Given Given nn(U) = 30, (U) = 30, nn(A) = 14, (A) = 14,

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

a)a) nn(A U B)(A U B)

b)b) nn(A, but not B)(A, but not B)

(b)(c)

(d)

(a)

A B

U

• b = 6

• a + b = 14

• b + c = 17

• a + b + c + d = 30

6

a + 6 = 14

a = 8 8

6 + c = 17 ; c = 11 11

8 + 6 + 11 + d = 30

25 + d = 30 ; d = 5 5

25

8

YOU DO:YOU DO:

Given Given nn(U) = 26, (U) = 26, nn(A) = 11, (A) = 11, nn(B) = (B) = 12 and 12 and nn(A B) = 8, find: (A B) = 8, find:

a)a) nn(A U B) (A U B)

b)b) nn(B, but not A)(B, but not A)

c)c) nn(A’)(A’)

A

(b)(c)

(d)

(a)

B

U

b = 8

a + b = 11; a = 3

b + c = 12; c = 4

a + b + c + d = 26

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

15

4

15

Now, the real thing… Now, the real thing…

A squash club has 27 members. 19 have black hair, 14 A squash club has 27 members. 19 have black hair, 14 have brown eyes and 11 have both black hair and have brown eyes and 11 have both black hair and brown eyes. brown eyes. – Place this information on a Venn DiagramPlace this information on a Venn Diagram– Find the number of members with:Find the number of members with:

Black hair or brown eyesBlack hair or brown eyes Black hair, but not brown eyesBlack hair, but not brown eyes

Black

(b)(c)

(d)

(a)

Brown

U

a + b + c + d = 27

a + b = 19

b + c = 14 c = 3

a = 8

d = 5

b = 11

YOU DO: YOU DO: Pele has 14 cavies as pets. Five have long hair Pele has 14 cavies as pets. Five have long hair

and 8 are brown. Two are both brown and and 8 are brown. Two are both brown and have long hair. have long hair.

a)a) Place this information on a Venn diagramPlace this information on a Venn diagram

b)b) Find the number of cavies that: Find the number of cavies that:

a)a) Are short hairedAre short haired

b)b) Have short hair and are brownHave short hair and are brown

c)c) Have short hair and are not brownHave short hair and are not brownLong

(b)(c)

(d)

(a)

Brown

U

a + b + c + d = 14

a + b = 5

b + c = 8

b = 2

c = 6

d = 3

a = 3

c + d = 9c = 6

d = 3

A little bit different… A little bit different…

A platform diving squad of 25 has 18 members A platform diving squad of 25 has 18 members who dive from 10 m and 17 who dive from 4 m. who dive from 10 m and 17 who dive from 4 m. How many dive from both platforms? How many dive from both platforms?

10 m

(b)(c)

(d)

(a)

4 m

U

1. a + b + c + d = 25

2. a + b = 18

3. b + c = 17

18 + c + 0 = 25

c = 7

Therefore b + 7 = 17 and b = 10

Now for the real real Now for the real real thing… thing…

dA B

CU

c ba

g

f

ed

U

c ba

g

f

bb

A city has three newspapers A, B, and C. Of the adult A city has three newspapers A, B, and C. Of the adult population, 1% read none of these newspapers, 36% read A, 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 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 C, 13% read A and C and 3% read all three papers. What percentage of the adult population read: percentage of the adult population read:

a)a) Newspaper A onlyNewspaper A only

b)b) Newspaper B or Newspaper CNewspaper B or Newspaper C

c)c) Newspaper A or B but not CNewspaper A or B but not C

U

f

b

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

HW #4 HW #4

4a – pg 78 #1; #4; pg 79 #7, 4a – pg 78 #1; #4; pg 79 #7,

pg 80 #9pg 80 #9 4b – pg 81, 82 #11 - 144b – pg 81, 82 #11 - 14