Relations and Functions


   
 
Types of Relations
Let R be a relation on a set A. Then R is said to be
 
 
 
 
 
 
 
 
 
Equivalence Relation
 
A relation which is reflexive, symmetric and transitive is called the Equivalence relation.
 
i.e., A relation R in a set A is called equivalence if it satisfies the following conditions.
 
 
 
 
Remarks:
 
The smallest equivalence relation in set A is the relation of equality in A.
 
The largest equivalence relation in A is A x A.
 
Inverse Relation
 
Let R Í A x B be a relation from A to B. Then the inverse of R, denoted by R-1 is a relation from B to A, defined as 
 
Clearly, Domain of R-1= Range of R and Range of R-1= Domain of R.
 
Example 1:
 
If R = {(1,2), (1,5),(2,4),(3,5)}
 
 
Domain of R-1= {2, 5, 4} = Range of R
 
Range of R-1= {1, 2, 3} = Domain of R
 
Example 2:
 
If f is a relation from a set A to set B such that f Í f-1 then, prove that f = f-1.
 
Suggested answer:
 
 
 
From (i) and (ii),
 
 
Some important relations
 
On the set of positive integers Z+, the relation “a divides b” is
 
reflexive
 
not symmetric
 
transitive
 
On the set of positive integers Z+, the relation "a
 
not reflexive
 
not symmetric
 
transitive
 
On the set of triangles, the relation “similar to” is
 
reflexive
 
symmetric
 
transitive
 
On the set of lines in a plane, the relation “parallel to” is
 
reflexive
 
symmetric
 
transitive
 
On the set of lines in a plane, the relation “perpendicular to” is
 
not reflexive
 
symmetric
 
not transitive
 
On the universal set of sets, the relation “subset of” is
 
reflexive
 
not symmetric
 
transitive
 
On the universal set of sets, the relation “subset of” is an
 
antisymmetric relation.
 
On the set of integers, the relation “congruence” is an equivalence relation.
 
 
     
   
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