What is binary relations and reflexive,symmetric and transitive.

Dear student! First of all ,I will tell you about the basic meaning of relation i.e It is a logical or natural association between two or more things; relevance of one to another; the relation between smoking and heart disease. The connection of people by blood or marriage. A person connected to another by blood or marriage; a relative. Or the way in which one person or thing is connected with another: the relation of parent to child. Now we turn to its mathematically definition, let A and B be any two sets. Then their cartesian product (or the product set) means a new set "A x B " which contains all the ordered pairs of the form (a,b) where a is in set A and b is in set B. Then if we take any subset say 'R' of "A x B" ,then 'R' is called the binary relation. Note All the subsets of the Cartesian product of two sets A and B are called the binary relations or simply a relation,and denoted by R. And note it that one raltion is also be the same as "A x B". Example: Let A={1,2,3} B={a,b} be any two sets. Then their Cartesian product means "A x B"={ (1,a),(1,b),(2,a),(2,b),(3,a),(3,b) } Then take any set which contains in "A x B" and denote it by 'R'. Let we take R={(2,b),(3,a),(3,b)} form "A x B". Clearly R is a subset of "A x B" so 'R' is called the binary relation. A reflexive relation defined on a set say ‘A’ means “all the ordered pairs in which 1st element is mapped or related to itself.” For example take a relation say R1= {(1,1), (1,2), (1,3), (2,2) ,(2,1), (3,1) (3,3)} from “A x B” defined on the set A={1,2,3}. Clearly R1 is reflexive because 1,2 and 3 are related to itself. A relation say R on a set A is symmetric if whenever aRb then bRa,that is ,if whenever (a,b) belongs to R then (b,a) belongs to R for all a,b belongs to A. For example given a relation which is R1={(1,1), (1,2), (1,3), (2,2) ,(2,1), (3,1) (3,3)} as defined on a set A={1,2,3} And a relation say R1 is symmetric if for every (a, b) belongs to R ,(b, a) also belongs to R. Here as (a, b)=(1,1) belongs to R then (b, a)=(1,1)also belongs to R. as (a,b)=(1,2) belongs to R then (b,a)=(2,1)also belongs to R. as (a,b)=(1,3) belongs to R then (b,a)=(3,1)also belongs to R.etc So clearly the above relation R is symmetric. And read the definition of transitive relation from the handouts and the book. You can easily understand it.