Numerical Sets
So what does this have to do with mathematics? When we define a set, all we have
to specify is a common characteristic. Who says we can't do so with numbers?
Set of Natural numbers: N = {1, 2, 3, 4, ...}
Set of entire numbers : J = {0 , 1 , 2 , 3 , 4 , 5 , ……}
Set of integers : I = {….. , -3 , -2 , -1 , 0 , 1 , 2 , 3 , …..}
Set of rational numbers: Q = { a/b : b ≠0 , a and b are real numbers }
Set of real numbers: R = {x : x is a real number }
Set of complex numbers: C  {z  x  i y, x, y  R , i  1}
SUBSETS
{1, 2, 3}  {1, 2, 3, 4, 5}
1  {1, 2, 3, 4, 5}
,
,
{1, 6}  {1, 2, 3, 4, 5}
6  {1, 2, 3, 4, 5}
EXAMPLE
X is the set of multiples of 3
Y is the set of multiples of 6
Z is the set of multiples of 9
Which one of the following is true? (⊂ means "subset")
A] X  Y
B] X  Z
C] Z  Y
D] Z  X
SOLUTIN
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X  3, 6, 9, 12,....
Y  6, 12, 18, 24,....
Z  9, 18, 27, 36,....
ANSWER: D] Z  X
EXAMPLE: Classify the different subsets of the following set X = { 1 , 2 , 3}
SOLUTION: 1] The empty set Φ
2] Subsets contains one element of X
{1},{2},{3}
3] Subsets contains two element of X
{ 1 , 2 } , { 1 , 3 } , { 2 , 3}
4] The non-proper set
X= { 1 , 2 , 3}
OPERATIONS ON SETS
X  Y  { a : a  X or a  Y }
X  Y  { a : a  X  a Y }
X  Y  { a : a  X and a  Y }
X  Y  { a : a  X  a Y }
THE DIFFERENCE BETWEEN TWO SETS
X  Y  { a : a  X and a  Y }
The symmetric difference between two sets
XY   X  Y   Y  X    X  Y    X  Y 
XY  { x : x   X  Y  or
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x  Y  X  }
XC  {a : a  U
COMPLEMENTARY SET
EXAMPLE:
If U  1, 2, 3, 4, 5 and Y  a : a2  8 a  15  0 . Find Y C
(a  3)(a  5)  0  a  3 , a  5
a 2  8 a  15  0
SOLUTION
Y  3, 5
, a  X}
⟹
Y C  U  Y  1, 2, 4
HOW WE PROVE SOME RELATIONS:
PROOF
x  (X  Y )  Z  x  (X  Y )   x  Z 
  x  X  x  Y    x  Z
  x  X    x  Y  x  Z
  x  X   Y  Z 
 x  X  Y  Z 
(X  Y )  Z  X  (Y  Z)
PROOF
(i) X  (Y  Z)  (X  Y )  (X  Z)
x  X  (Y  Z)  x  ( X )   x  Y  Z 
  x  X    x  Y  x  Z
  x  X  x  Y    x  X  x  Z
  x  X  Y    x  X  Z
 x  X  Y    X  Z 
 X  (Y  Z)  (X  Y )  (X  Z)
(V) De’ Morgan Laws
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PROOF
(i) (X  Y )C  XC  Y C
x  ( X  Y )C  x  U  x  ( X  Y )
 x U  x  X  x U  x Y 
 x U  X  x U  Y 
  x  XC    x  Y C 
  x  XC  Y C 
 ( X  Y )C  X C  Y C
EXAMPLE : If the universal set is given by U={1,2,3,4,5,6}, and A={1,2}
,B={2,4,5}, C={1,5,6} are three sets, find the following sets:
a] A  B
,
b] A  B
c] AC  B
,
d ] A  BC
e] Check De Morgan's law by finding
 A  B   AC  B C
C
 A  B   AC  BC
C
SOLUTION
U={1,2,3,4,5,6}, and A={1,2} , B={2,4,5}, C ={1,5,6}
AC  3, 4,5,6 , BC  1,3,6
a] A  B  1, 2, 4,5 ,
b] A  B  2
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c] AC  B  2,3, 4,5, 6
d ] A  BC  1, 2
e]  A  B   3,6
C
(i)
AC  BC   3, 4,5,6   1,3,6  3,6
(ii)
From (i) , (ii) we get
 A  B
C
 AC  BC
Also,
 A  B
C
 2  1,3, 4,5,6
C
(iii)
AC  BC  3, 4,5,6  1,3,6  1,3, 4,5,6
From (iii) , (iv) we get
 A  B
C
 AC  BC
PROVE (HOMEWORK):
(X  Y )  Z  X  (Y  Z)
X  Y  Z    X  Y    X  Z 
 X  Y C  X C  Y C 
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(iv)