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Properties of Relation in a Set Reflexive Relations: Let R be a relation in a set A. then R is called a reflexive relation if(a, a) Î...

Aptitude for GATE , PSU Exams - Relations



Properties of Relation in a Set
Reflexive Relations: Let R be a relation in a set A. then R is called a reflexive relation if(a, a)ÎR for all aÎA.
          In other words, R is reflexive if every element in A is related to itself. Thus, R is reflexive if aRa holds for all aÎA.
          As relation R is a set A is not reflexive if there is atleast one element such that (a, a)ÏR.
Ex: Let A={1, 2, 3, 4}. Then the relation
R1={(1, 1), (2, 4), (3, 3), (4, 1), (4, 4)} in A is not reflexive since 2ÎA but (2, 2)ÏR1.
Anti-Reflexive Relations: Let R be a relation in a set A. Then R is called an anit-reflexive relation if (a, a)ÏR for every aÎA. i.e., a/Ra for all aÎA.
Ex: The relation in the set of natural numbers N defined by
(i)          ‘x<y’ is anti-reflexive, since a is not less than a for any natural number a,
(ii)         Let A={1, 2, 3, 4}. Then the relation Ri={(1, 1), (2, 3), (3, 4)} is not anti-reflexive since (1, 1)ÎR.
Symmetric Relations:  Let R be a relation in a set A. Then R is said to be symmetric relation if
(a, b)ÎR Þ (b, a)ÎR.
          Thus R is symmetric if bRa holds whenever aRb holds.
          A relation R is set A is not symmetric if there exist two distinct elements a, bÎA, such that aRb, but b/Ra.
Ex: Let R be the set of all striaght lines in aplane. The relation R is defined by ‘x is parallel to y’ is symmetric, since if a straight line a is parallelto a straight line b, then b is also parallel to a, thus
          (a, b)ÎR Þ (b,a) ÎR
Anti-Symmetric Relation:  Let R be a relation in a set A, i.e., let R be a subset of A ´ A. Then R is said to be an anti-symmetric relation if
          (a, b)ÎR and (b, a)ÎR Þa=b
          Thus R is anti-symmetric  if a≠b then possibly (a, b)ÎR or possibly (b, a)ÎR, be never both.
          A relation R is a set A is not anti-symmetric if there exist elements a, bÎA, a≠b such that (a, b)ÎR and (b, a)ÏR.
Ex:  Let A be a family of sets and let R be the relation in A defined by ‘x is subset of y’ then R is anti-symmetric since AÍ B and BÍA ÞA=B.
Transitive Relations:  Let R be a relation in a set, i.e., let R be a subset of A´A then R is said to be a transitive relation if
          (a, b)ÎR and (b, c)ÎR Þ(a, c)ÎR.
          A relation R in a set is not transitive if there exist elements a, b, cÎa, not necessarily distinct, such that (a, b)ÎR, (b, c)ÎR but (a, c)ÏR.
Ex: Let L be the set of all straight lines in a plane and R be the relation in L defined by ‘x’ is parallel to y’. If a is parallel to b and b is parallel to c then obviously a is parallel to c. Thus (a, b)ÎR and (b,c)ÎRÞ(b,c)ÎR. Hence R is transitive.
Equivalence Relations:
Let R be a relation in a set A. Then R is an equivalence relation in A if and only if
(i)          R is reflexive, i.e., for all aÎR, (a, a)Îr.
(ii)         R is symmetric, i.e., (a, b)ÎR Þ(b,a)ÎR, for all a, bÎA.
(iii)        R is transitive, i.e., (a, b)ÎR and (b, c)ÎR Þ(a, c)ÎR, for all a, b, cÎA.
Ex: (i) The most trivial example of an equivalence relation is that of ‘equality’ for any elements in any set.
I.            A=a, i.e., reflexive
II.          a=bÞ b=a i.e., symmetric
III.        a=b and b=cÞa=c, i.e., transitive.
(ii) Let R be the relation in the real numbers defined by ‘x£y’. Then
I.            x £ x i.e.(x,x)ÎR, i.e., R is reflexive.
II.          Let x£ y but y not £ x i.e., (x, y)ÎR but not necessarily, (y, x)ÎR i.e., R is not symmetric. Thus R is not an equivalent relation.

Partial Ordered Relation:  A relation ‘R’ on A such that ‘R’ is reflexive, anti symmetric and transitive is called partial order relation on ‘A’.
Note:  
(i)          Union of ‘2’ Reflexive or symmetric relation is a reflexive or symmetric relation.
(ii)         Union of ‘2’ transitive relations need not be transtitive.

(iii)        Intersection of ‘2’ equivalance relation is an equivalence relation.