Types of Relations

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  • Q1

    Let X be a family of sets and R be a relation on X defined by 'A is disjoint from B'. Then R is

    Marks:1
    Answer:

    symmetric.

    Explanation:
    Clearly, the relation is symmetric but it is neither reflexive nor transitive.

     

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  • Q2

    Let R = {(1, 3), (4, 2), (2, 4), (2, 3), (3, 1)} be a relation on the set A = {1, 2, 3, 4}. The relation R is

    Marks:1
    Answer:

    not symmetric.

    Explanation:

    (2, 3)  R but (3, 2)  R

     R is not symmetric

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  • Q3

    Let R be a relation on a set A such that R = R-1, then R is

    Marks:1
    Answer:

    symmetric.

    Explanation:
    Let (a, b)  R. Then,
    (a, b)  R  (b, a)  R-1   [by defn of R-1]
     (b, a)  R                       [ R = R-1]
    So, R is symmetric.
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  • Q4

    Let R = {(2, 3), (3, 4)} be a relation defined on the set A = {1, 2, 3, 4}. The minimum number of ordered pairs required to be added in R so that the new relation becomes an equivalence relation is

    Marks:1
    Answer:

    8

    Explanation:

    Given R = {(2,3), (3,4)}             ...(i)

    To make it reflexive, we add the following ordered pairs in R:

    R = {1, 1), (2, 2), (3, 3), (4, 4), (2, 3), (3,4) ...(ii)

    Hence 4 ordered pairs are added.

    To make it symmetric, we again add the following ordered pairs in R from eqn (ii)

    {(1,1), (2,2), (3,3), (4,4), (2,3) (3,2), (3,4) (4,3)} ...(iii)

    Hence two more ordered pairs are added.

    Finally to make it transitive, we add the following ordered pairs in R from eqn (iii)

    {(1,1), (2,2), (3,3), (4,4), (2,3) (3,2), (3,4) (4,3), (2,4), (4,2)}

    Hence, two more ordered pairs are added.

     

    Total 8 ordered pairs must be added to make the relation R an equivalence relation.

     

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  • Q5

    For real number x any y, we write xRy x-y+2 is an irrational number. Then the relation R is

    Marks:1
    Answer:

    reflexive only

    Explanation:

    We have xR, x -x+2=2, an irrational number.

    Hence, xRx x, i.e. R is reflexive.

    R cannot be symmetric because if x = 2, y = 1, then

    x-y+2=22-1, an irrational number, i.e. , xRy.

    But y-x+2=1-2+2=1, a rational number.  

    Thus, xRy  yRx

    Similarly, we may show that R is not transitive.

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