Follow Us


Question 15

Let R be the relation in the set {1, 2, 3, 4} given by R = {(1, 2), (2, 2), (1, 1), (4, 4), (1, 3), (3, 3), (3, 2)}. Choose the correct answer.
(A) R is reflexive and symmetric but not transitive.
(B) R is reflexive and transitive but not symmetric.
(C) R is symmetric and transitive but not reflexive.
(D) R is an equivalence relation.

Answer

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

It is seen that (a, a) ∈ R, for every a ∈{1, 2, 3, 4}.

∴ R is reflexive.

It is seen that (1, 2) ∈ R, but (2, 1) ∉ R.

∴R is not symmetric.

Also, it is observed that (a, b), (b, c) ∈ R ⇒ (a, c) ∈ R for all a, b, c ∈ {1, 2, 3, 4}.

∴ R is transitive.

Hence, R is reflexive and transitive but not symmetric.

The correct answer is B.

Popular Questions of Class 12th mathematics

 

">

Let A = R – {3} and B = R – {1}. Consider the function  f : A → B defined by

. Is f one-one and onto? Justify your answer. 

 

  • Q:- Show that the relation R in the set A = {1, 2, 3, 4, 5} given by R = { (a,b) ; |a - b| is even}, is an equivalence relation. Show that all the elements of {1, 3, 5} are related to each other and all the elements of {2, 4} are related to each other. But no element of {1, 3, 5} is related to any element of {2, 4}.
  • Q:-

     Prove that the Greatest Integer Function f : R → R, given by f(x) = [x], is neither one-one nor onto, where [x] denotes the greatest integer less than or equal to x.

  • Q:- Show that the relation R in the set R of real numbers, defined as R = {(a, b): a ≤ b2} is neither reflexive nor symmetric nor transitive.
  • Q:-

     Show that the Modulus Function f : R → R, given by f(x) = |x|, is neither oneone nor onto, where | x | is x, if x is positive or 0 and |x| is – x, if x is negative.

  •  

    ">

    Let A = R – {3} and B = R – {1}. Consider the function  f : A → B defined by

    . Is f one-one and onto? Justify your answer. 

     

  • Q:-

     Let f : {1, 3, 4} → {1, 2, 5} and g : {1, 2, 5} → {1, 3} be given by f = {(1, 2), (3, 5), (4, 1)} and g = {(1, 3), (2, 3), (5, 1)}. Write down gof. 

  • Q:- Determine whether each of the following relations are reflexive, symmetric and transitive:
    (i) Relation R in the set A = {1, 2, 3,13, 14} defined as
    R = {(x, y): 3x − y = 0}
    (ii) Relation R in the set N of natural numbers defined as
    R = {(x, y): y = x + 5 and x < 4}
    (iii) Relation R in the set A = {1, 2, 3, 4, 5, 6} as
    R = {(x, y): y is divisible by x}
    (iv) Relation R in the set Z of all integers defined as
    R = {(x, y): x − y is as integer}
    (v) Relation R in the set A of human beings in a town at a particular time given by
    (a) R = {(x, y): x and y work at the same place}
    (b) R = {(x, y): x and y live in the same locality}
    (c) R = {(x, y): x is exactly 7 cm taller than y}
    (d) R = {(x, y): x is wife of y}
    (e) R = {(x, y): x is father of y}
  • Q:-

    State with reason whether following functions have inverse

    (i) f : {1, 2, 3, 4} → {10} with

    f  = {(1, 10), (2, 10), (3, 10), (4, 10)}

    (ii) g : {5, 6, 7, 8} → {1, 2, 3, 4} with

    g = {(5, 4), (6, 3), (7, 4), (8, 2)}

    (iii) h : {2, 3, 4, 5} → {7, 9, 11, 13} with

    h = {(2, 7), (3, 9), (4, 11), (5, 13)}

     

  • Q:- Check whether the relation R in R defined as R = {(a, b): a ≤ b3} is reflexive, symmetric or transitive.
  • Write a Comment: