This document discusses relations and their properties. It defines a relation as a subset of A x B where A and B are sets. Relations can exist between a set and itself. Common examples of relations are subsets representing inequalities or functions. The properties of relations that are discussed include reflexivity, symmetry, antisymmetry, and transitivity. Graphs can be interpreted as relations based on their edge sets. Relations are useful in theoretical computer science, game theory, and logic.

1. Submitted to:- Submitted by:-
Dr. Madan Mohan Sati Garishma Bhatia
B.Tech – III sem
Seminal Presentation
Topic:- Relations

2.  Relations are useful when studying properties of
things.There are plenty of areas in theoretical
computer science that require the use of
relations.
 In fact, a graph can be looked as a way of
interpreting relations.This is so because the edge
set is a subset ofV×V.
 Relations are also useful in topics like game
theory and logic.

3.  Let A & B be sets.
 A binary relation from A to B is a
subset of A x B.
 Let R be a relation. If ( a, b )  R, we
write a R b.

4.  A relation on a set A is a relation fromA to A.
 Examples of relations on R:
 R1 = { (a, b) | a  b }.
 R2 = { (a, b) | b = +sqrt( a ) }.
 Are R1 & R2 functions?

5. A relation R onA is:
 Reflexive: a ( aRa ).
Are either R1 or R2 reflexive?
 Symmetric: a b ( aRb  bRa ).
 Let S be a set of people.
 Let R &T be relations on S,
R = { (a, b) | a is a sibling of b }.
T = { (a, b) | a is a brother of b }.
Is R symmetric?
IsT symmetric?

6.  Antisymmetric:
1. a b ( ( aRb  bRa )  ( a = b ) ).
2. a b ( ( a  b )  ( ( a, b )  R  ( b, a )  R ) ).
Example: L = { ( a, b ) | a  b }.
Can a relation be symmetric & antisymmetric?
 Transitive:
a b c ( ( aRb  bRc )  aRc ).
Are any of the previous examples transitive?

7.  It is useful studying properties of thing.
 There are plenty of areas in theoritical computer science
that require the use of relations.
 In fact, a graph can be looked as away of interpreting
relations.This is so because the edge set is a subset of
V×V.
 Relations are also useful in topics like game theory and
logic.

8. THANK
YOU