Review of the basics
Relation.
A relation between sets A and B is a subset “~” of A  B .
( a, b)  ~ is read as “ a is related to b ” and write a ~ b . (Note: ( a, b) is
an ordered pair.)
If A  B  S , ~ is a relation on S .
Function.
A function f mapping X into Y (denoted by f : X  Y ) is a
relation between X and Y with the property that x  X , ! ( x, y )  f
where y  Y .
X : domain
Y : codomain
f ( X )  { f ( x ) | x  X } is the range of f .
☆  : X  Y is one to one if x1, x2  X ,  ( x1 )   ( x2 )  x1  x2 .
(1-1) (injection)
Note.  is a function mapping X into Y if x1  x2   ( x1 )   ( x2 ) and
x  X , x is related to some y  Y .
f : A  B , B '  f ( A) , f 1 ( B ')  {x | f ( x)  B '} .
☆  : X  Y is onto if  ( X )  Y .
(surjection)
1–1 + onto = bijection
1-1
(＊) f : X 
Y , f 1 ( y )  x (where f ( x )  y ) defines the inverse
οntο
function of f .
| A | : cardinality of A .
Definition. Two sets X and Y have the same cardinality if there exists
a bijection between X and Y .
| Z |  | Z |   0
How much do you know about cardinality?
Theorem.
11
f : A 
 B | A |  | B | .
f : A 

 B | A |  | B | .
onto
Theorem. | A |  | p( A) | .
11
Proof. x , let f ( x )  {x} , f : A 
 p( A) . Hence | A |  | p( A) | . Suppose
that | A |  | p( A) | . Then,  : A  p( A) which is a bijection.
Consider x  A , either x   ( x ) or x   ( x ) .
Let B  {x  A | x   ( x )} . Since  is 1-1 and onto, there exists an
element b  A s.t.  (b)  B . By the def. of B ,
b   (b)  B  b  B  .
Equivalence relation.
Def. An equivalence relation ~ on a set is a relation on S s.t. x, y, z  S .
1. x ~ x . (Reflexive)
2. x ~ y  y ~ x . (Symmetric)
3. x ~ y, y ~ z  x ~ z . (Transitive)
Partition. A partition of a set, P (S )  {S1, S2 , } such that S is a disjoint
union of S1, S2 , . Each Si is called a “cell” of the partition.
(Partition may be infinite.).
Theorem. Let S be a nonempty set and ~ be an equivalence relation on
S . Then ~ yields a partition of S where a  {x  S | x ~ a, a  S} .
Also, each partition of S where a ~ b iff a and b are in
the same cell of the partition.
Proof. Let a  {x  S | x ~ a, a  S} .
Claim a  b or a  b   .
(1) a  S , a  a . (Reflexive)
(2) a  b    a  b .
let c  a  b
c ~ a and c ~ b .
 (symmetric)
a~c
x  a  x ~ a  x ~ c (Transitive)
x~b
 x b
a b
Similarly, b  a .
Def. (Congruence Modulo n )
Let n  Z  . The relation “  n ” defined by “ a  n b  def n | a  b ” is an
■
equivalence relation a  n b is also written as a  b (mod n ). The cells obtained by
applying the equivalence relation  n to Z is called the residue classes modulo n
in Z .
We can exted the residue classes modulo n in “ Z ” to “ Z ”.
Example. The residue classes modulo 2 in Z are 2 and 1 where
2  { 2 , 4 , 6 , and
} 1  {1,3,5, } .
Extend to Z , the residue classes modulo 2 in Z are
0 {
, 6 , 4, 2 , 0 , 2 , 4and
, 6 , 1 }{ , 5, 3, 1,1,3,5, } .
{ 0 , 1}Z2