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Types of Relations on the Set A

Let A be a set and R is a relation on A, i.e. R A × A. Then we define

  1. Void Relation : If R = φ , then R is called a void relation on A.
  2. Universal Relation : If R = A×A, then R is called an universal relation on A.
  3. Identity Relation : A relation R is defined as an identity relation if R = {(a, a) : a  A}. Thus in an identity relation on A, every element of A is related to itself only. Identity relation on A is also denoted by IA. Thus
    IA = {(a, a) : a A}
    Example If A = {1, 2, 3}, then IA = {(1, 1), (2, 2), (3,3)}
  4. Reflexive Relation : A relation R is said to be a reflexive relation on A if every element of A is related to itself.
    Thus R is reflexive (a,a) R, i.e. aRa711.pngaA
    [The symbol716.pngis read as “for every element”]
    Example : Let A = {1, 2, 3} be a set.
    Then R = {(1, 1), (2, 2), (3, 3), (1, 3), (2, 1)} is a reflexive relation on A.
  5. Symmetric Relation : A relation R on a set A is defined as a symmetric relation if (a,b) R (b, a) R That is, aRb bRa, where a, bA.
    Example : Let A = {1, 2, 3, 4} and let R1 be relation on
    A given by R1 = {(1, 3), (1, 4), (3, 1), (2, 2), (4, 1)} is a symmetric relation on A.
  6. Transitive relation : A relation R on a set A is defined as a transitive relation if (a,b) R and (b,c) R (a,c) R
    That is, aRb and bRc aRc, where a, b, c, A.
    Example : Let L be the set of all straight line in a plane. Then the relation ‘is parallel to on L is a transitive relation, because of any 123 L.
    1||2 and 2||3  1||3
  7. Antisymmetric Relation : A relation R on a set A is antisymmetric if (a,b) R and (b,a) R a = b
    If (a, b)  R and (b, a) R, then still R is an antisymmetric relation.
    Example : Let R be a relation on the set N of natural numbers defined by
    xRY  ‘x divides y’ for all x, y  N
    This relation is a antisymmetric relation on N.
    Since for any two numbers a, b  N.
    a/b and b/a  a = b i.e. aRb and bRa  a = b

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