Cambridge International Mathematics

Sets (Chapter 2) 67

2aList:

i setA ii setB iii setU

iv setA\B v setA[B

b Find:

i n(A) ii n(B) iii n(U)

iv n(A\B) v n(A[B)

3 Consider U=fxjx 612 ,x 2 Z+g,

A=f 2 , 7 , 9 , 10 , 11 g and B=f 1 , 2 , 9 , 11 , 12 g.

a Show these sets on a Venn diagram.

b List the elements of: i A\B ii A[B iii B^0

c Find: i n(A) ii n(B^0 ) iii n(A\B) iv n(A[B)

4 IfAis the set of all factors of 36 andBis the set of all factors of 63 , find:

a A\B b A[B

5 If X=fA, B, D, M, N, P, R, T, Zg and Y=fB, C, M, T, W, Zg, find:

a X\Y b X[Y

6 Suppose U=fxjx 630 ,x 2 Z+g,

A=ffactors of 30 g and B=fprime numbers 630 g.

a Find: i n(A) ii n(B) iii n(A\B) iv n(A[B)

b Useato verify that n(A[B)=n(A)+n(B)¡n(A\B)

7aUse the Venn diagram given to show that:

n(A[B)=n(A)+n(B)¡n(A\B).

b SupposeAandBare disjoint events.

Explain why n(A[B)=n(A)+n(B):

8 Simplify:

a X\Y for X=f 1 , 3 , 5 , 7 g and Y=f 2 , 4 , 6 , 8 g

b A[A^0 for any set A 2 U.

c A\A^0 for any set A 2 U.

USING VENN DIAGRAMS TO ILLUSTRATE REGIONS

We can use a Venn diagram to help illustrate regions such as the union or intersection of sets.

Shaded regions of a Venn diagram can be used to verifyset identities. These are equations involving sets

which are true forallsets.

Examples of set identities include:

A[A^0 =UA\A^0 =?

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

U

A

B^8

5 3

1

6

4

2

7

()

()=+

a

a

nA a b:

means that

there are elements

in this region, so

¡

¡¡¡¡

¡

U

AB

()a

()b

()c

()d

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Y:\HAESE\IGCSE01\IG01_02\067IGCSE01_02.CDR Thursday, 11 September 2008 11:13:44 AM PETER