Prof. Xiaorong Gan
Kunming University of Science & Technology
ch0
An Introduction to Set Theory
--- Some Basic Definitions and Results
Sets
Subsets
Operations on sets
Further results
0.1 sets
A set is any well-defined collection of distinct objects.
Some special sets are denoted by symbols.For example,
R denotes the set of all real numbers, Z denotes the set
of all integers while Z+ denotes the set of all positive
integers.
The objects in a set are called the elements or members
of the set. We write x  S to denote the fact that the
fact that the object x is an elements of the set S. If x is
not a member of S then we write x  S
0.2 Describing a set
(i) Listing the elements of a set : we denote the set by
enclosing all its elements in a list within curly brackets.
e.g.
S  {1,3,5,7} is the statements that S is the set
whose four elements are 1, 3, 5 and 7.
(ii) The conditional definition of a set : a set may be
described by specifying some condition which
determines whether or not an object is an element of
the set.
e.g.
S  {n  Z 0  n  6
 means that S is the set of
(ii) The conditional definition of a set : a set may be
described by specifying some condition which
determines whether or not an object is an element of
the set.
e.g. S  {n  Z 0  n  6  means that S is the set of
all integers n which satisfy 0  n  6.
We could have described this particular set by listing
the elements. i.e. S  {1,2,3,4,5}.
(iii) The constructive definition of a set: here we give a
formula for constructing the elements of the set.
e.g. S  {n2 | n  Z is the set of integer squares which
(iii) The constructive definition of a set: here we give a
formula for constructing the elements of the set.
e.g. S  {n2 | n  Z is the set of integer squares which
may be listed as 0,1,4,9,16
A set S is said to be finite if it contains only finitely many
different members; otherwise it is said to be an infinite
set.If S is a finite set then the number of different objects
in S is called the order (or cardinality) of S and is
denoted by | S | .
0.3 Equality pf sets
Two sets are said to be equal if they have precisely the
same elements. i.e. C=D Means x  C if and only if (iff)
xD
To show that two sets C and D are equal it is necessary to
prove two things (although this can often be done
together in simple cases): prove that every element of C
is an element of D and conversely that every element of
D is an element of C.
0.4 Subsets
Given two sets A and B, we say that A is subset of B,
written A  B , when every element of A is also an
element of B.
i.e. If x  A , Then x  B. Thus, A=B if and only if
A  B and B  A.
If A and B are in fact unequal so that B contains at least
some element not contained in A, then we say that A is a
proper subset of B and this is denoted by A  B.
If we have sets A, B, C with A  B and B  C , then A  C .
If we have sets A, B, C with A  B and B  C , then A  C .
Note that when all sets being considered are subsets of
a particular set S then S is called the universal set.
The empty set is the unique set which has no elements at
all and is denoted by the symbol Ø.
e.g. the set x  R x 2  1 is clearly empty.
The complement of a subset: Let A be a subset of a
given set S. The complement of A in S is the subset
C
{ x  S | x  A} and is denoted by A ( A )
0.5 Operations on sets
Let A and B be two subsets of a universal set S.
(i) The intersection of A and B is the subset of S
comprising those elements which belong to both A
and B. Thus A  B  x  S | x  A and x  B
A and B are said to be disjoint if A  B  
i.e. A and B have no elements in common at all.
(ii) The union of A and B is the subset of S comprising
those elements which belong to A or to B or to both.
Thus
A  B  x  S | x  A or x  B
(ii) The union of A and B is the subset of S comprising
those elements which belong to A or to B or to both.
Thus
A  B  x  S | x  A or x  B
The above two definitions are easily extended to the
case when we are considering the intersection or union
of n>2 sets.
i.e. let A1 , A2 , Ann be n subsets of S. Then, the
intersection  Ai comprises those elements in S
i 1
that belong to all of A1 , A2 ,, An
n
i.e.
A
i
i 1
 { x  S | x  Ai for all i  1,2,, n}
n
i.e.
A
i
 { x  S | x  Ai for all i  1,2,, n}
i 1
n
while the union
A
i
comprises those elements in S that
i 1
belong to at least one of the subsets A1 , A2 , An .
n
i.e.
A
i
 { x  S | x  Ai for at least one i  1,2,, n}.
i 1
(iii) The difference of A and B, denoted A – B, consists of
elements which lie in A but not in B. Thus
A  B  { x  S | x  A and x  B}
0.6 Further results
Let A, B and C be subsets of some universal set S , then
we have the following identities:
(i) Commutatively:
A  B  B  A and A  B  B  A.
(ii) Associativity:
A  ( B  C )  ( A  B )  C and A  ( B  C )  ( A  B )  C .
(iii) Distributivity: A  ( B  C )  ( A  B )  ( A  C ) and
A  ( B  C )  ( A  B )  ( A  C ).
C
C
(iv) Complementation: A  A  S and A  A  
(v) Double complement: ( AC )C  A.
(i) Commutatively: A  B  B  A and A  B  B  A.
(ii) Associativity:
A  ( B  C )  ( A  B )  C and A  ( B  C )  ( A  B )  C .
(iii) Distributivity: A  ( B  C )  ( A  B )  ( A  C ) and
A  ( B  C )  ( A  B )  ( A  C ).
C
C
(iv) Complementation: A  A  S and A  A  
(v) Double complement: ( AC )C  A.
(vi) de Morgan’s laws:
( A  B)C  AC  BC and ( A  B)C  AC  BC .
Example :Consider a fair dice. The dice is six-sided and
is numbered 1 through to 6. Now this dice is rolled.
Example :Consider a fair dice. The dice is six-sided and
is numbered 1 through to 6. Now this dice is rolled.
Let A=odd number , B= number less than 5 and C=even
number but less than 5 . Find the following sets:
(a) S= 1,2,3,4,5,6 (b) A= 1,3,5
(c) B= 1,2,3,4
(d) C= 2,4
(e) A  B  1,2,3,4,5
(f) A  B  1,3 (g) A C  
(h) A  B  5
(i) B  A  2,4
(j) Ac  B 1,2,3,4,6
A
A B
B
A
A B
B
A
B
BB A
B  A( A  B)
A  B( A  B   )
A
B
A  B( A  B   )
A
A ( A )
C
A