Class XII ISC Maths Chapter Relation and Functions Exercise 1.1
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class 12 relation&function exercise1-1
Relations & Functions
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Q1 Determine whether each of the following relations are reflexive, symmetric and transitive :
(i) Relation R in the set A = {1, 9, 3, ..., 101} defined by R = {(x, y) :2x — y = 0}.
(ii) Relation R in the set Z of all integers defined by
R = {(x,y) : x — y is an integer}
Q2 If the relation R in the set A, where A = {1, 2, 3, 4, 5, 6}, is defined by R = {(x, y) : y is divisible by x}, then express R in the roster form. Also determine whether the relation R is
(i) reflexive
(ii) symmetric
(iii) transitive
Q3 If R is the relation defined on the set of natural numbers N as follows:
R = {(x, y); x, y ∈ N, 2x + y = 41}, find the domain and the range of the relation R.
Determine whether the relation is reflexive, symmetric and transitive.
Q4 Determine whether each of the following relations in the set A of human beings in a city at a particular time are reflexive, symmetric and, transitive
(i) R = {(x, y) : x and y work at the same place}
(ii) R = {(x, y) : x and y live in the same locality}
(iii) R = {(x, y) : x is exactly 7 cm taller than y}
(iv) R = {(x, y) : x and y live within 2 kilometres}
Q5 Determine whether each of the following relations in the set A of students at a particular time are symmetric and transitive:
(i) R = {(x, y) : x, y ∈ A, x and y are honest)
(ii) R = {(x, y): x , y ∈ A, x and y are obedient}
(iii) R = {(x, y) : x, y ∈ A, x and y are hardworking)
What are the advantages of students being honest, obedient and hardworking?
Q6 Show that the relation R in the set A = {1, 2, 3} given by R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 3)} is reflexive but neither symmetric nor transitive.
Q7 Determine whether each of the following relation are reflexive , symmetric and transitive
(i) R = {(a, b) : a, b ∈ N, a < b}
(ii) R = {(a, b) : a, b ∈ R, a ≤ b }
Q8 Let T be the set of all triangles drawn in a plane with R as a relation in T given by R = {(T1, T2) : T1 ≅ T2}. Show that R is an equivalence relation.
Q9 Show that the relation R in the set A of all books in a library of a school, given by R = {(x,y) : x and y have same number of pages} , is an equivalence relation.
Q10 Show that the relation R on the set I of all integers defined by R = {(a, b) : a - b is divisible by 3, a, b ∈ I}, is an equivalence relation.
Q11 Prove that the relation R in the set A = {5, 6, 7, 8, 9), given by R = {(a, b) : |a - b| is divisible by 2}, is an equivalence relation. Find all elements related to element
Q12 Give examples of relations which are
(i) symmetric but neither reflexive nor transitive.
(ii) transitive but neither reflexive nor symmetric.
(iii) symmetric and transitive but not reflexive.
(iv) symmetric and reflexive but not transitive.
(v) reflexive and transitive but not symmetric.
Q13 Give an example of a relation R on A = {a, b, c} which is
(i) neither reflexive nor symmetric but transitive.
(ii) neither symmetric nor transitive but reflexive.
(iii) neither transitive nor reflexive but symmetric.
Q14 Show that the relation S in the set A ={x ∈ Z : 0 ≤x ≤12 } S ={(a, b) : a, b ∈ A, |a—b| is divisible by 4} Find the set of elements related to 1.
Q15 Show that the relation R in the set A = {x ∈ W , 0≤x≤17} given by
R={(a,b) : |a — b| is a multiple of 5}
(ii) R = {(a, b) : a = b}
are equivalence relations. Find the set of all elements related to 2 in each case.
Q17 (i) If R = {(x, y) : x + 2y = 8} is a relation on N, write the range of R.
(ii) Let R = {(a, a3) : a is a prime number less than 5}. Find the range of R.
Q18 Let A be any non-empty set. State true or false :
(i) Identity relation on A is reflexive.
(ii) Every reflexive relation on A is identity relation on A.
(iii) Identity relation on A is symmetric.
(iv) Identity relation on A is an equivalence relation.
(v) Universal relation on A is an equivalence relation.
Q19 State the reason for the relation R in the set {1, 2, 3} given by R = {(1, 2), (2, 1)} not to be transitive.
Q21 If A = {0, 1, 2, ..., 9} and the relation R on A is defined by R = {(x, y) : x, y ∈ A, y = 2x + 1}, then determine whether the relation R is
(i) reflexive (ii) symmetric (iii) transitive.
Q22 Is the relation R on the set R of real numbers defined by R = {(a, b) : a, b ∈ R, 1 + ab ≥ 0} transitive? Justify your answer.
Q23 Is the relation R on the set Q of rational numbers defined by R = {(x, y) : x, y ∈ Q, x < y 2}, symmetric? Justify your answer.
Q24 If the relation R on the set A = {1, 2, 3} is defined by R = {(1, 1), (2, 2),(3,3),(2,1),(3,2)} then determine whether the relation R is
(i) reflexive (ii) symmetric (iii) transitive.
Q25 If 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)}, then determine whether
(i) R is reflexive and symmetric but not transitive.
(ii) R is reflexive and transitive but not symmetric.
(iii) R is symmetric and transitive but not reflexive.
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