Let A and B be two non-empty sets. Then the relation which is usually denoted by R from A to B is defined as the set of ordered pairs (in each of which the first element is from A and the second element is from B) satisfying a certain condition.
The set of all first elements of ordered pairs of R is called “Domain” of R.
The set of all second elements of ordered pairs is called “range” of R.
Example:
If A = {1,4,6}, B= {1, 2, 3} then construct a relation R from A to B describing “is a multiple of”.
Since 1 “is amultiple of” 1, (1,1) is in R.
Since 4 “is a multiple of” 2 (4,2) is in R.
Similarly we can find all the ordered pairs of R.
Hence R = {(1,1)(4,1)(4,2)(6,1)(6,2)(6,3)}
Here, Domain of R = {1,4,6} and
Range of R = {1,2,3}
Note : Though (4,3) is an ordered pair, it won’t be there in R because 4 is not a multiple of 3.
Graphical representation of a relation:
A relation is graphically represented as follows.
If A = {1,4,6}, B= {1, 2, 3} then a relation R = {(1,1)(4,2)(6,2)(6,3)} from A to B is represented through a graph as follows.
Types of relations:
The types of relations are nothing but their properties. There are different types of relations namely reflexive, symmetric, transitive and anti symmetric which are defined and explained as follows through real life examples.
Reflexive relation:
A relation R is said to be reflexive over a set A if (a,a) € R for every a € R.
Example:
If A is the set of all males in a family, then the relation “is brother of” is not reflexive over A. Because any person from the set A cannot be brother of himself.
Symmetric relation:
A relation R is said to be symmetric if (a,b) € R => (b,a) € R
Example:
If A is the set of all males in a family, then the relation “is brother of” is symmetric over A.
Because if a is brother of b then b is brother of a.
Transitive relation:
A relation R is said to be symmetric if (a, b) € R, (b, c) € R => (a, c) € R.
Example:
If A is the set of all males in a family, then the relation “is brother of” is transitive over A.
Because if a is brother of b and b is brother of c then a is brother of c.
Anti symmetric relation:
A relation R is said to be anti symmetric if (a,b) € R and (b, a) € R => a=b.
Example:
The relation “≤” on the set of real numbers is anti symmetric because if a ≤ b and b ≤ a then a=b.
SchoolTutoring Academy is the premier educational services company for K-12 and college students. We offer tutoring programs for students in K-12, AP classes, and college. To learn more about how we help parents and students in Oxnard visit: Tutoring in Oxnard.