Cambridge International Mathematics

Sets (Chapter 2) 69

4 The Venn diagram alongside is the most general case for three events in

the same sample spaceU. On separate Venn diagrams shade:

a A b B^0 c B\C

d A[C e A\B\C f (A[B)\C

g (A[C)\B h (A\C)[B i (A[B)^0 \C

5 Verify that:

a (A\B)^0 =A^0 [B^0

b A[(B\C)=(A[B)\(A[C)

c A\(B[C)=(A\B)[(A\C)

When we solve problems with Venn diagrams, we generally do not deal with individuals. Instead, we simply

record thenumberof individuals in each region.

Example 11 Self Tutor

The Venn diagram alongside illustrates the number of people in

a sporting club who play tennis (T) and hockey (H).

Determine the number of people:

a in the club b who play hockey

c who play both sports d who play neither sport

e who play at least one sport.

a Number in the club= 15 + 27 + 26 + 7 = 75 b Number who play hockey=27+26=53

c Number who play both sports=27 d Number who play neither sport=7

e Number who play at least one sport= 15 + 27 + 26 = 68

Example 12 Self Tutor

In a class of 24 boys, 16 play football and 11 play baseball. If two play neither game, how many play

both games?

Method 1:

Letxbe the number who play both games.

) 16 ¡x play football and 11 ¡x play baseball.

2 boys play neither sport.

) (16¡x)+x+ (11¡x)+2=24

) 29 ¡x=24

) x=5

So, 5 play both games.

F PROBLEM SOLVING [9.3, 9.4]

DEMO

U

T H

() 15 () 27 () 26

() 7

U

FB

() 2

()16¡-¡x ¡()x ()11¡-¡x

U

A B

C

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Y:\HAESE\IGCSE01\IG01_02\069IGCSE01_02.CDR Monday, 15 September 2008 12:07:24 PM PETER