♨ |
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, anti-symmetric as well as
transitive is called an Partial Ordered Relation. |
♦ |
Relation Matrix: In a binary relation, matrix elements mij
= 1 iff aiRaj holds good, otherwise mij
= 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 & x2 + y2 =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. |
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Working- Put 2 sets of same cardinality. write the relations.
click 1. Then click the image of any relation and then click-3.
repeat this for all images. The images u have finished will be
yellow. |
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