Cambridge International Mathematics

58 Sets (Chapter 2)

Theelementsof a set are the objects or members which make up the set.

We use 2 to mean ‘is an element of’or‘is a member of’

and 2 = to mean ‘is not an element of’or‘is not a member of’.

So, for P=f 2 , 3 , 5 , 7 g we can write 22 P and 42 =P:

There are 4 elements in the setP, so we write n(P)=4.

n(A) reads ‘the number of elements in setA’.

SUBSETS

The elements of the set S=f 2 , 5 , 7 g are also elements of the set P=f 2 , 3 , 5 , 7 g:

We say thatSis asubsetofP, and write SμP:

Ais asubsetofBif all elements ofAare also elements ofB.

We write AμB:

For example, f 1 , 3 gμf 1 , 2 , 3 g but f 1 , 2 , 3 g * f 1 , 3 g.

Two setsAandBareequalif their elements are exactly the same.

For example, f 2 , 3 , 5 , 7 g=f 5 , 3 , 7 , 2 g

THE UNIVERSAL SET

Associated with any set is a universal set, denoted U.

Theuniversal setUcontains all of the elements under consideration.

For example, if we are considering the positive integers less than 20 , then

U=f 1 , 2 , 3 , 4 , 5 , 6 , 7 , ...... , 19 g.

These dots indicate the continuation of the pattern up to the final element.

InU, the set of prime numbers is P=f 2 , 3 , 5 , 7 , 11 , 13 , 17 , 19 g and the set of composite numbers is

Q=f 4 , 6 , 8 , 9 , 10 , 12 , 14 , 15 , 16 , 18 g.

Notice that PμU and QμU:

THE EMPTY SET

Sometimes we find that a set has no elements. Such a set is called theempty set, and is denoted?orfg.

The empty set?is a proper subset of any other set.

Ais aproper subsetofBif every element ofAis also an element ofB, but A 6 =B.

We write A½B:

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Y:\HAESE\IGCSE01\IG01_02\058IGCSE01_02.CDR Monday, 15 September 2008 12:03:57 PM PETER