RELATIONS
We have several relations in our family life such as father and the son father and the daughter, brother and brother, brother and sister etc. Similar relations exist between two objects such as people, number, ideas etc.
For example : Rama was the son of Dashratha, Sita was the wife of Rama etc. In fact a relation is a set of ordered pairs.
Ordered pair : An ordered pair (x,y) is a well defined pair of things which exists in a particular order, ‘x’ is called first coordinate and y is called the second coordinate.
Two ordered pairs of a numbers equal when both the coordinates are equal. If n(A) = a and n(B) = b then number of ordered pairs is (a, b).
Some points about Relations :
- An ordered pair shown by a point is known as a lattice
- A relation between a set A and Set B is a subset of AC
- The association of a pair of quantities is called a
- A relation should have some
Showing a Relation : Following are the ways of showing a relation.
- Roaster form : In this a set of ordered pairs is described.
e.g. (1) Sri Lanka, Colombo; England, London, India, Delhi, France, Paris.
- {5, -5}, {2, -2}, {7, -7} In this form we record a set of ordered pairs, which satisfy a given
For example :
- {(Colombo, Sri Lanka), (London, England), (Tokyo, Japan), (New Delhi, India)} Here R mean “is the capital of”.
(2) {(5, -5), (2, -2), (7, -7)}. Here R mean “is the negative of”.
- Set Builder Form : A rule is used for description of a set of ordered pairs, in the form {(a, b) Î A C B : (a…b)}, the blank is to be replaced by the
For example :
(1) Let R = {(2,4), (4,8), (8,16)} when A = {2,4,8} and B = {4,8,16} Clearly R =
{(a,b) Î A C B : x is half of b}
- By Arrow Diagrams : In this the relation is shown by arrows drawn from the first component to the
For example : Set A “is the capital of’ set B. New Delhi—– India
Tokyo—— Japan
London—— England
Colombo——- Sri Lanka
These can be shown as set of ordered pairs {(Colombo, Sri Lanka), (London, England), (Tokyo, Japan), (New Delhi, India)}
Properties of relations reflexive : A relation on a set A is called reflexive, if every element has a relation with itself. i.e., xRx, for all x Î A. Thus, it contains ordered pairs (x,x) whose first component is the same as the second.
Symmetric : A relation on a set A is called symmetric, if aRb and bRa are such that aRb Þ bRa i.e., (a,b) Î R Þ b (b,a) Î R. Thus if x is related to b, the b is related to a.
Transitive : A relation R on a set A is called transitiive if aRb and bRa Þ aRc.
Equivalence relation : A relation that is reflexive, symmetric and transitive is known as an equivalence relation. Thus a relation R on a set A is an equivalence rela- tion if it satisfies the following conditions :
- aRb for all a Î A
- aRb Þ bRa; ab Î A
- aRb and bRa Þ aRc; a,b,c Î A
