EXAMPLE OF QUESTION~~

Let M = {1,2,3,4,5,6,7,8,9,10,11,12,13,14,15}. A = {2,4,7,10,13,15},  B = {3,4,7,10,15}, C = {3,4,8,14} and D = {3,4,7,9} Determine :

i) A Ụ B        ii) A ∩ D      iii) (A Ụ B) ∩ C     iv) B /C     v) B + C


Answer:


i) A Ụ B = {2,3,4,7,10,13,15}
ii) A ∩ D = {7}
iii) (A Ụ B) ∩ C = {3,4}
iv) B /C = {2}
v) B + C = {4,5,7,9,10,15)

TYPES OF RELATION

Reflexive Relation

    Definition: A relation R on a set X is called reflexive if (x,x) ϵ R for every x ϵ X.

    Example :

  Let R be the relation on  X = {1, 2, 3,4, 5 } defined by (x, y) ϵ R if x  ≤  y  for x,y ϵ X.

  List R.  Is R reflexive ?

    Answer:

      R= { (1,1), (1,2), (1,3), (1,4), (1,5), (2,2), (2,3), (2,4), (2,5), (3,3), (3,4), (3,5), (4,4), (4,5), (5,5) }.

      Reflexive = (1,1),(2,2),(3,3),(4,4),(5,5)

      Therefore, R is reflexive.

Symmetric relation

       Definition : A relation R on a set X is called symmetric if for all x,y ϵ X , then (y,x) ϵ R.

       Example  :

        Consider the relation R on X = { a, b, c, d } given as follows

          R = { (a,d), (b,c), (c,b), (d,a ) }. Is R symmetric? Why?

       Answer :

         Symmetric = (1,3),(3,1)

           R is symmetric. Because all its elements is in R.

Antisymmetric relation

       Definition : A relation R on a set X is called antisymmetric if for all x,y ϵ X, if  (x,y) ϵ R and 

                       x ≠ y then (y,x) ϵ R.

       Example  :

         Consider the relation R on X = { a, b, c, d } given as follows

          R = { (a,b), (b,c), (c,d), }. Is R symmetric or antisymmetric ? Why ?

        Answer :

          R is antisymmetric . Because   all x,y ϵ X, if  (x,y) ϵ R and x ≠ y then (y,x) ϵ R.

Transitive relation

       Definition : A relation R on a set X is called transitive if for all x, y, z  ϵ X, if (x,y) and (y,z) ϵ R 

                        then (x,z) ϵ R.

       Example:

         Consider the relation R on X = { a, b, c, d } given as follows

          R = { (a,b), (a,d), (a,c), (b,c), (b,d), (c,d) }. Is R transitive? Why?

       Answer :

         It is transitive. Because all x, y, z  ϵ X, if (x,y) and (y,z) ϵ R then (x,z) ϵ R.

           i.e (a,b) ϵ R, ( b,c) ϵ R , then (a,c) ϵ R  also.

Equivalence Relation

Definition: A relation that is reflexive, symmetric and transitive on set X is called an equivalence relation on X.

Example:

Consider the relation R on {1, 2, 3, 4,5 } defined as

R = { (1,1), (1,2), (1,3), (1,5), (2,2), (2,4), (2,5), (3,1), (3,3), (3,5), (4,2), (4,4), (5,1) , (5,3), (5,5)}.

Justify that R is equivalence relation.

Answer:

  R is reflexive because { (1,1), (2,2), (3,3), (4,4), (5,5)} ϵ R.

  R is symmetric because whenever  (x,y) ϵ R then (y,x,) ϵ R.

  R is transitive  because whenever  (x,y) ϵ R and   (y,z) ϵ R then  (x,z) ϵ R.

 Therefore R is an equivalence relation on {1, 2, 3, 4}.