Cambridge Additional Mathematics

Sets and Venn diagrams (Chapter 1) 13

COUNTING ELEMENTS OF SETS

The number of elements in setAis written n(A).

For example, the set A=f 2 , 3 , 5 , 8 , 13 , 21 g has 6 elements, so we write n(A)=6.

A set which has a finite number of elements is called afinite set.

For example: A=f 2 , 3 , 5 , 8 , 13 , 21 g is a finite set.

?is also a finite set, since n(?)=0.

Infinite setsare sets which have infinitely many elements.

For example, the set of positive integersf 1 , 2 , 3 , 4 , ....g does not have a largest element, but rather keeps

on going forever. It is therefore an infinite set.

In fact, the sets N,Z,Z+,Z¡,Q, andR are all infinite sets.

SUBSETS

SupposeAandBare two sets.Ais asubsetofBif every element

ofAis also an element ofB. We write AμB.

For example, f 2 , 3 , 5 gμf 1 , 2 , 3 , 4 , 5 , 6 g as every element in the first set is also in the second set.

Ais aproper subsetofBifAis a subset ofBbut isnot equal toB.

We write A½B.

For example, Z½Q since any integer n=

n

1

2 Q. However,^122 Q but^122 =Z.

We use A*B to indicate thatAisnota subset ofB

and A 6 ½B to indicate thatAisnota proper subset ofB.

UNION AND INTERSECTION

IfAandBare two sets, then

² A\B is theintersectionofAandB, and consists of

all elements which are inbothAandB

² A[B is theunionofAandB, and consists of all

elements which are inAorB.

For example:

² IfA=f 1 , 3 , 4 g and B=f 2 , 3 , 5 g then A\B=f 3 g

and A[B=f 1 , 2 , 3 , 4 , 5 g.

² The set of integers is made up of the set of negative integers, zero,

and the set of positive integers: Z=(Z¡[f 0 g[Z+)

DEMO

Every element in

and every element in

is found in.

A

B

AB[

4037 Cambridge

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