Download chapter 3 part 1:sets

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PART I –SETS
1.1 SETS
1.1.1 DEFINITION OF SET
1.1.2 METHODS FOR SPECIFYING SET
1.1.3 SUBSETS
1.1.4 VENN DIAGRAM
1.1.5 SET OPERATIONS
1.1.6 SET IDENTITIES
1.1.1 DEFINITION OF SET:
 Unordered collection of distinct objects and may be
viewed as any well-defined collection of objects
called elements or members of the set .
Notations:
 Usually uses capital letters, A, B, X, Y to denote sets.
 lowercase letters, a, b, x, y to denote elements of
sets.
 x  A - denote x is an element of set A.
 x  A - denote x is not an element of set A.
  or 
 - empty set
If G is the set of all even numbers, then :
4 G
3G
Special Symbols:
 N = the set of natural numbers or positive integers:
1, 2, 3,...
 Z = the set of all integers: ..., -2, -1, 0, 1, 2,...
 Q = the set of rational numbers
 R = the set of real numbers
 C = the set of complex numbers
3.1.2 METHODS FOR SPECIFYING SETS:
1) Listing: by listing its elements between curly
brackets { } and separating them by commas.
E.g: A = {0}, B = {2, 67, 9}, C = {x, y, z}.
2) Set’s Construction / Implicit Description: by giving
a rule which determines if a given object is in the
set or not.
E.g: A = { x : x is a natural number}
B = { x : x is an even integer, x > 0}
C = { x : 2x = 4}
 We describe a set by listing its element only if the set
contains a few elements; otherwise we describe a set by
the property which characterizes its element.

List all the elements of each set when N = {1, 2, 3,
...}.
i)
ii)
iii)
A   x   3  x  9
B   x   x is even, x  11
C   x   4  x  3
3.1.3 SUBSETS:
 If every element of A is also an element of B.
 That is a  A, a  B
 Written as A  B
Two sets are equal if they both have the same
elements, or equivalently if each is contained in the
other. That is:
A  B if and only if A  B and B  A
 If A is not a subset of B, or at least
one element of A
does not belong to B, we write A  B .
Subsets:
 Property 1: It is common practice in mathematics to
put a vertical line “|” or slanted line”/” through a
symbol to indicate the opposite or negative meaning
of a symbol.
 Property 2: The statement A  B does not mean
the possibility that A  B . In fact, for every set A we
have A  A since every element in A belongs to A.
However, if A  B and A  B , then we say A is a
proper subset of B (sometimes written A  B ).
 Property 3: Suppose every element of a set A
belongs to a set B and every element of B belongs to
a set C. Then clearly every element of A also belongs
to C. In other words, if A  B and B  C , then A  C.
Let A = {2, 3, 4, 5},
 a) Show that A is not a subset of B   x   x is even
 b) Show that A is a proper subset of C  1, 2,3,...,8,9
x A
3.1.4 VENN DIAGRAMS:
 Pictorial representation of set in which sets are
represented by enclosed areas in the plane.
 A large square is used to represent the universe set
U.
 The element of the universe in a set S fall inside the
circle for S, while elements not in the set S fall
outside of that circle.
U
S
x A
1)


3.1.5 SETS OPERATIONS :
Symmetric
Union
Difference
Intersection
Complement
Disjoint
Difference
Union
Let A and B be sets. The union of sets A and B
contain those elements that are either in A or B, or
in both ( A  B   x x  A or x  B ).
Denoted: A  B
U
x A
A
B
Find the union of the sets A  1,3,5 and B  1, 2,3.
x A
2)


Intersection
Let A and B be sets. The intersection of sets A and B
contain those elements in both A and B
( A  B   x x  A and x  B ).
Denoted: A  B
U
A
x A
B
Find the intersection of the sets;
A  1,3,5 and B  1, 2,3
x A
Properties of Union and Intersection

Property 1: Every element x in belongs to both A
and B; hence x belongs to A and x belongs to B. Thus
A  B is a subset of A and B; namely
A  B  A and A  B  B

Property 2: An element x belongs to the union A  B
if x belongs to A or x belongs to B; hence every
element in A belongs to A  B , and every element in
B belongs to A  B . That is,
x A
A  A  B and B  A  B
3)

Disjoint
Two sets are called disjoint if their intersection is
the empty set ( A  B   ).
x A
Suppose sets A  1, 2 , B  4,5,6 and C  5,6,7,8 .
Find A  B, A  C and B  C .
x A
4)

Difference
Let A and B be sets. The difference of A and B is the
set containing those elements in Abut not in B .
A  B   x x  A and x  B

Denoted by A  B
U
A

B
Also called the complement of B with respect to A.
x A
Find the difference of 1,3,5 and 1, 2,3.
5)

Complement
Let U be the universal set. The complement of the
set A is the complement of A with respect to U .
A   x x  U and x  A

Denoted by A / Ac / A'
U
A

Similarly can be define as U  A .
x A
Suppose U = 1, 2,3,... is the universal set. Let A  1, 2,3, 4 , B  3, 4,5,6,7
C  2,3,8,9 , E  2, 4,6,... E is the set of even integers  .
Find:
a) A, B, and E
b) A  B, B  C , and E  A
6)


Symmetric Difference
The symmetric difference of sets A and B consists
of those elements which belong to A or B but not
to both
A  B   x x  A and x  B or  x  B and x  A
Denoted by A  B .
U
A
x A
B
Suppose U = 1, 2,3,... is the universal set. Let A  1, 2,3, 4 , B  3, 4,5,6,7
C  2,3,8,9 , E  2, 4,6,... E is the set of even integers  .
Find:
a) A  B
b) B  C
c) A  C
3.1.6 SET IDENTITIES:
Sets under operations of union, intersection, and
complement satisfy various laws / identities which are
listed in Table 1.
x A
x A
1
2
3 ) In a survey of 260 college students, the following data were obtained:
 64 had taken a mathematics course,
 94 had taken a computer science course,
 58 had taken a business course,
 28 had taken both mathematics and business course,
 26 had taken both mathematics and computer science course,
 22 had taken both a computer science and a business course, and
 14 had taken all three types of courses.
(a) How many students were surveyed who had taken none of the three
types of courses?
106
(b) Of the students surveyed, how many had taken only a computer
science course?
60