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

U B A

AB U

AB 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}

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

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

Four regions on a Venn Diagram A B U 1)In A, but not in B 2)In B, but not in A 3)In both A and B 4)Neither in A nor B

Example 1 Given the Venn Diagram below, how many elements are there in Given the Venn Diagram below, how many elements are there in a) P b) P U Q c) Q’ d) P, but not Q e) Q, but not P f) neither P nor Q? P (3) (11) (4) (7) Q U

YOU DO: Give the number of elements in: Give the number of elements in: a) X’ b) X Y c) X U Y d) X, but not Y e) Y, but not X f) Neither X nor Y X (6) (3) (2) (8) Y U

Example 2: Given n(U) = 30, n(A) = 14, Given n(U) = 30, n(A) = 14, n(B) = 17 and n(A B) = 6 find: a) n(A U B) b) n(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 = c = 17 ; c = d = d = 30 ; d =

YOU DO: Given n(U) = 26, n(A) = 11, n(B) = 12 and n(A B) = 8, find: Given n(U) = 26, n(A) = 11, n(B) = 12 and n(A B) = 8, find: a) n(A U B) b) n(B, but not A) c) n(A’) A (b) (c) (d) (a) B U b = 8 a + b = 11; a = 3 b + c = 12; c = 4 a + b + c + d = d = 26; d =

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. 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 or brown eyes Black hair, but not brown eyes Black 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: Pele has 14 cavies as pets. Five have long hair and 8 are brown. Two are both brown and have long hair. a) Place this information on a Venn diagram b) Find the number of cavies that: a)Are short haired b)Have short hair and are brown c)Have short hair and are not brown Long (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 = 9 c = 6 d = 3

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 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? 10 m (b) (c) (d) (a) 4 m U 1.a + b + c + d = 25 2.a + b = 18 3.b + c = c + 0 = 25 c = 7 Therefore b + 7 = 17 and b = 10

Now for the real real thing… d AB C U c b a g f e d U c b a g f bb 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: 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: a) Newspaper A only b) Newspaper B or Newspaper C c) Newspaper 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 = 36; therefore g = 36 e = 40; therefore e = 24 f = 32; f = 31 and h = 1

HW #4 4a – pg 78 #1; #4; pg 79 #7, 4a – pg 78 #1; #4; pg 79 #7, pg 80 #9 4b – pg 81, 82 # b – pg 81, 82 #