♨ 
Relation R is a subset of a cartesian product of A ⊗ B i.e.
R⊄ AxB . R' which is a subset of BxA is called an
inverse relation denoted by R^{1} 
‡ 
Example: Set A={1,2,3} and Set B ={1,3,5} Then K=AxB
={(1,1),(1,3),(1,5),(2,1),(2,3),(2,5),(3,1),(3,3),(3,5)} 
↹ 
Set M ={(1,1),(1,3),(3,1),(3,3)} is subset of K and therefore a
relation of A →B 
➰ 
If a Relation is defined between two sets, it is called a Binary
Relation. 
➲ 
A relation R is said to be Reflexive if aRa holds true for
∀a ⊄ M . M is a reflexive relation because of
(1,1) & (3,3) 
➲ 
A relation R is said to be symmetric if aRb
⇒ bRa ∀a,b ⊄ M. Here, (1,3) & (3,1) both
belong to M. M is a symmetric relation because of (1,1) & (3,1) 
♦ 
A relation is said to be transitive if aRb, bRc ⇒
aRc ∀a,b,c ⊄ M . Here M is not a transitive relation. 
♦ 
A relation which is reflexive, symmetric as well as transitive
is called an Equivalent Relation. 
♦ 
A relation which is reflexive, antisymmetric as well as
transitive is called an Partial Ordered Relation. 
♦ 
Relation Matrix: In a binary relation, matrix elements m_{ij}
= 1 iff a_{i}Ra_{j} holds good, otherwise m_{ij}
= 0. If R ⊄ AxA, then relation matrix is a square matrix
of the same order as that of A. Otherwise, it is a rectangular
matrix of m rows and n columns where m is the order of A and n is
the order of B. 
♦ 
Example: Suppose R consists of ordered set of integers (x,y)
such that (x,y) ∈ R & x^{2} + y^{2} =25 .
Then
R={(5,0),(0,5),(0.5),(5,0),(3,4),(4,3),(3,4),(4,3),(4,3),(3,4),(3,4),(4,3)}
and A= {5,4,3,0,3,4,5} . Order of A is 7 and therefore no. of
elements of relation matrix will be 49. 
♦ 
The diagonal elements m11,m22,m33,m4,m55,m66,m77 are equal to
zero. m14=m41=1, m23=m32=1, m25=m52=1,m36=m63=1,
m47=m74=1,m56=m65=1. Rest are all zero. Here R=R^{1} 
♦ 
It is a symmetric matrix if both set A and B
have the same cardinality, otherwise it is a rectangular matrix. 

Working Put 2 sets of same cardinality. write the relations.
click 1. Then click the image of any relation and then click3.
repeat this for all images. The images u have finished will be
yellow. 

