Exercise 1.1
Question 1
Determine whether each of the following relations are reflexive, symmetric and transitive :
(i) Relation R in the set A = {1, 2, 3, …, 10} 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}
(iii) Relation R in the set N of all natural numbers defined by
R = {(x, y) : y = x + 5, x < 4}.
Sol :
Question 2
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.
Sol :
Question 3
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.
Sol :
Question 4
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 5 cm taller than y}.
(iv) R = {(x, y) : x and y live within 2 kilometres}.
(v) R = {(x, y) : x is wife of y}.
Sol :
Question 5
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?
Sol :
Question 6
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.
Sol :
Question 7
Decide in each of the following cases whether the relation is symmetric, transitive and reflexive. Justify your answer by giving examples.
(i) ‘Is less than’ on N
(ii) ‘Is a factor of’ on N
Sol :
Question 8
Let T be the set of all triangles drawn in a plane with R as a relation in T given by $R = {(T_1, T_2) : T_1 ≅ T_2}$. Show that R is an equivalence relation.
Sol :
Question 9
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.
Sol :
Question 10
Show that the relation R in the set A of points in a plane, given by R = {(P, Q) : points P and Q have equal distances from the origin}, is an equivalence relation. Also show that the set of all points related to a point P (different from origin) is the circle passing through P with origin as its centre.
Sol :
Question 11
Show that the relation R in the set A of all triangles, given by $R = {(T_1, T_2) : T_1 ~ T_2}$, is an equivalence relation. Consider three triangles $T_1$ with sides 3, 4, 5; $T_2$ with sides 5, 12, 13 and $T_3$ with sides 6, 8, 10. Which triangles among $T_1$, $T_2$ and $T3_$ are related?
Sol :
Question 12
Show that the relation R in the set A of all polygons, given by $R = {(P_1, P_2) : P_1$ and $P_2$ have same number of sides}, is an equivalence relation. What is the set of all elements in A related to the right triangle T with sides 3, 4 and 5?
Sol :
Question 13
Show that the relation R defined in the set L of all straight lines drawn in the XY-plane, given by $R = {(L_1, L_2) : L_1 \text{ is parallel to }L_2}$, is an equivalence relation. Find the set of all straight lines related to the line y = 2x + 4.
Sol :
Question 14
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
Sol :
Question 15
Let I be the set of all integers and R be the relation on I defined by R = {(a, b) : a – b is divisible by 5}. Prove that R is an equivalence relation. Find the set of all elements of I related to 1.
Sol :
Question 16
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.
Sol :
Question 17
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.
Sol :
Question 18
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. Also 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}.
Sol :
Question 19
Show that the relation S in the set A = {x ∈ Z : 0 ≤ x ≤ 12} given by
S = {(a, b) : a, b ∈ A, | a – b | is divisible by 4} is an equivalence relation.
Find the set of elements related to 1
Sol :
Question 20
Show that the relation R in the set A = {x ∈ W, 0 ≤ x ≤ 17} given by
(i) 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.
Sol :
Question 21
Consider the division of set A = {1, 2, 3, 4, 5, 6, 7, 8} by subsets {1, 6}, {2, 7}, {3, 8}, {4} and {5}. Show that the relation R in A, given by R = {(a, b) : a and b lie in the same subset of the division of A}, is an equivalence relation. Find the set of all elements of A related to 6 and the set of all elements related to 5.
Sol :
Very Short Answer Type Question
Question 22
If A = {– 1, 1, 3}, then what is the number of relations on A?
Sol :
Question 23
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
Sol :
Question 24
State the reason for the relation R in the set {1, 2, 3} given by R = {(1, 2), (2, 1)} not to be transitive.
Sol :
Question 25
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
Sol :
Question 26
Let R be the relation on the set N given by R = {(a, b) : a = b – 2, b > 6}, then determine whether
(i) (2, 4) ∈ R
(ii) (6, 8) ∈ R
(iii) (9, 7) ∈ R.
Sol :
Question 27
If the relation R on the set N of natural numbers is defined by
R = {(a, b) : a, b ∈ N, a = 2b – 1, b > 4}, then determine whether
(i) (6, 11) ∈ R
(ii) (13, 7) ∈ R
(iii) (5, 3) ∈ R
Sol :
Question 28
If P be the set of people living in Delhi and R be the relation on P defined by
R = {(a, b) : a, b ∈ P, a lives within 4 km of b}, then determine whether R is transitive.
Sol :
Question 29
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.
Sol :
Question 30
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
Sol :
Question 31
If A = {1, 3, 7} and R be the relation ‘is greater than’ on the set A. Write R as a set of ordered pairs. Is this relation an equivalence relation?
Sol :
Question 32
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
Sol :
Question 33
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
Sol :
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