RELATIONS AND FUNCTIONS

Part III - Binary operation
1) Definition
2) Properties of Binary operation a) Commutative b) Associative c) Existence
of identity (identity element) d) Existence of inverse (inverse of a element)
3) Number of Binary operation on a set RELATIONS I. Cartesian product 1. A
B (x, y) / x A and y B       (a, b) (c,d)  (a, b) (c,d) a c and b d  
  If A B   then either one of A or B is null set 2. If n (A) = m and n (B)
= n then n(A×B) = mn 3. If   2 n(A B) m then n (A B) (B A) m     
 4. A B B A(in general)but A B B A A B         where A  
and B  5. A (B C) (A B) (A C)       and A (B C) (A B) (A C) 
     and A (B C) (A B) (A C)       6. (A B) (C D) (A C) (B
D
I. Relation
1. A relation from A to B is a subset of A ×B, If (x, y)  R means x related to y.
ie., if xRy then (x, y) R . ie R = (x, y) / x A, y B    If n(A) = m and n(B)
= n then number relations from A to B = 2mn 2. Inverse relation If 1 R :A B
then R : B A      1 R (y, x) /(x, y) R    Note : Dom (R) = Ran
(R-1) and Ran (R) = Dom(R-1) 3. Relation on a set R :A A  is the relation on
a set A No. of relation on a set have n elements = 2 n 2 II. Types of relations
1. Reflexive If a R a  a A 2. Symmetric If a R b  b R a   a, b A 3.
Transitive If a R b and b R c    aRc a,b,c A 4. Equivalence relation If R
is equivalence then it is reflexive, symmetric and transitive. 5. Identity relation
- IA : AA Let A be a set then the relation IA = (x, y) / x A, y A and x y  
  is called identity relation 6. Inverse relation If 1 R : A B then R : B A  
   1 R (y, x) /(x, y) R    7. Void relation :- Let A be any set   
 A A, is called the void relation 8. Universal relation Let A be any set then A
A A A    , A × A is called the universal relation III. Properties on relations
Let R1 and R2 be two relations on a set A 1. If R R 1 2  and R1 is reflexive
then R2 is reflexive. 2. If R1or R2 is reflexive then R R 1 2 is reflexive 3. If
R1 and R2 is reflexive then R R 1 2 is reflexive 4. If R1 and R2 are
symmetric then 1 1 R , R , R R , R R ,R R 1 2 1 2 1 2 1 2 and R2 -
R1 are symmetric 5. If R1 and R2 are transitive then R1 R2 is transitive
but R1 R2 is need not be transitive 6. If R1 and R2 are two equivalence
relation then R1 R2 is need not be an equivalence relation. But R1 R2 is
equivalence relation. 7. No. of possible relation from O(A).O(B) A B 2 