Number Theory: An Introduction to Mathematics

2 I The Expanding Universe of Numbers

We will not formally define aset, but will simply say that it is a collection of

objects, which are called itselements. We writea∈Aifais an element of the setA

anda∈/Aif it is not.

A set may be specified by listing its elements. For example,A={a,b,c}is the set

whose elements area,b,c. A set may also be specified by characterizing its elements.

For example,

A={x∈R:x^2 < 2 }

is the set of all real numbersxsuch thatx^2 <2.

If two setsA,Bhave precisely the same elements, we say that they areequaland

writeA=B. (IfAandBare not equal, we writeA=B.) For example,

{x∈R:x^2 = 1 }={ 1 ,− 1 }.

Just as it is convenient to admit 0 as a number, so it is convenient to admit the
empty set∅, which has no elements, as a set.
If every element of a setAis also an element of a setBwe say thatAis asubset
ofB,orthatAisincludedinB,orthatB contains A, and we writeA⊆B.Wesay
thatAis apropersubset ofB, and writeA⊂B,ifA⊆BandA=B.
Thus∅⊆Afor every setAand∅⊂AifA=∅. Set inclusion has the following
obvious properties:
(i) A⊆A;
(ii)if A⊆B and B⊆A,then A=B;
(iii)if A⊆B and B⊆C,then A⊆C.

For any setsA,B,the set whose elements are the elements ofAorB(or both) is
called theunionor ‘join’ ofAandBand is denoted byA∪B:

A∪B={x:x∈Aorx∈B}.

The set whose elements are the common elements ofAandBis called theintersection
or ‘meet’ ofAandBand is denoted byA∩B:

A∩B={x:x∈Aandx∈B}.

IfA∩B=∅,thesetsAandBare said to bedisjoint.

A

B

A

B

A∪BA∩B

Fig. 1.Union and Intersection.