RD Sharma solutions for Mathematics [English] Class 12 chapter 1 - Relations [Latest edition]

Let A be the set of all human beings in a town at a particular time. Determine whether the following relation is reflexive, symmetric and transitive:

 R = {(x, y) : x and y work at the same place}

Let A be the set of all human beings in a town at a particular time. Determine whether the following relation is reflexive, symmetric and transitive:

R = {(x, y) : x and y live in the same locality}

Let A be the set of all human beings in a town at a particular time. Determine whether the following relation is reflexive, symmetric and transitive:

R = {(x, y) : x is wife of y}

Let A be the set of all human beings in a town at a particular time. Determine whether the following relation is reflexive, symmetric and transitive:

R = {(x, y) : x is father of y}

Three relations R1, R2 and R3 are defined on a set A = {a, b, c} as follows:
R1 = {(a, a), (a, b), (a, c), (b, b), (b, c), (c, a), (c, b), (c, c)}
R2 = {(a, a)}
R3 = {(b, c)}
R4 = {(a, b), (b, c), (c, a)}.

Find whether or not each of the relations R1, R2, R3, R4 on A is (i) reflexive (ii) symmetric and (iii) transitive.

Test whether the following relation R₁ is  (i) reflexive (ii) symmetric and (iii) transitive :

R₁ on Q₀ defined by (a, b) ∈ R₁ ⇔ a = 1/b.

Test whether the following relation R₂ is (i) reflexive (ii) symmetric and (iii) transitive:

R₂ on Z defined by (a, b) ∈ R₂ ⇔ |a – b| ≤ 5

Test whether the following relation R₃ is (i) reflexive (ii) symmetric and (iii) transitive:

R₃ on R is defined by (a, b) ∈ R₃ `⇔` a² – 4ab + 3b² = 0.

Let A = {1, 2, 3}, and let R₁ = {(1, 1), (1, 3), (3, 1), (2, 2), (2, 1), (3, 3)}, R₂ = {(2, 2), (3, 1), (1, 3)}, R₃ = {(1, 3), (3, 3)}. Find whether or not each of the relations R₁, R₂, R₃ on A is (i) reflexive (ii) symmetric (iii) transitive.

The following relation is defined on the set of real numbers.
aRb if a – b > 0

Find whether relation is reflexive, symmetric or transitive.

The following relation is defined on the set of real numbers.

aRb if 1 + ab > 0

Find whether relation is reflexive, symmetric or transitive.

The following relation is defined on the set of real numbers.  aRb if |a| ≤ b

Find whether relation is reflexive, symmetric or transitive.

Check whether the relation R defined in the set {1, 2, 3, 4, 5, 6} as R = {(a, b) : b = a + 1} is reflexive, symmetric or transitive.

Check whether the relation R in R defined by R = {(a, b) : a ≤ b³} is reflexive, symmetric or transitive.

Prove that every identity relation on a set is reflexive, but the converse is not necessarily true.

If A = {1, 2, 3, 4} define relations on A which have properties of being reflexive, transitive but not symmetric ?

If A = {1, 2, 3, 4} define relations on A which have properties of being symmetric but neither reflexive nor transitive ?

If A = {1, 2, 3, 4} define relations on A which have properties of being reflexive, symmetric and transitive ?

Let R be a relation defined on the set of natural numbers N as
R = {(x, y) : x, y ∈ N, 2x + y = 41}
Find the domain and range of R. Also, verify whether R is (i) reflexive, (ii) symmetric (iii) transitive.

Is it true that every relation which is symmetric and transitive is also reflexive? Give reasons.

An integer m is said to be related to another integer n if m is a multiple of n. Check if the relation is symmetric, reflexive and transitive.

Show that the relation '≥' on the set R of all real numbers is reflexive and transitive but not symmetric ?

Give an example of a relation which is reflexive and symmetric but not transitive?

Give an example of a relation which is reflexive and transitive but not symmetric?

Give an example of a relation which is symmetric and transitive but not reflexive?

Give an example of a relation which is symmetric but neither reflexive nor transitive?

Give an example of a relation which is transitive but neither reflexive nor symmetric?

Given the relation R = {(1, 2), (2, 3)} on the set A = {1, 2, 3}, add a minimum number of ordered pairs so that the enlarged relation is symmeteric, transitive and reflexive.

Let A = {1, 2, 3} and R = {(1, 2), (1, 1), (2, 3)} be a relation on A. What minimum number of ordered pairs may be added to R so that it may become a transitive relation on A.

Let A = {a, b, c} and the relation R be defined on A as follows: R = {(a, a), (b, c), (a, b)}. Then, write minimum number of ordered pairs to be added in R to make it reflexive and transitive.

Defines a relation on N :
  x > y, x, y ∈  N

Determine the above relation is reflexive, symmetric and transitive.

Defines a relation on N :

x + y = 10, x, y∈ N

Determine the above relation is reflexive, symmetric and transitive.

Defines a relation on N:

xy is square of an integer, x, y ∈ N

Determine the above relation is reflexive, symmetric and transitive.

Defines a relation on N:

x + 4y = 10, x, y ∈ N

Determine the above relation is reflexive, symmetric and transitive.

Show that the relation R defined by R = {(a, b) : a – b is divisible by 3; a, b ∈ Z} is an equivalence relation.

Show that the relation R on the set Z of integers, given by
R = {(a, b) : 2 divides a – b},  is an equivalence relation.

Prove that the relation R on Z defined by
(a, b) ∈ R ⇔ a − b is divisible by 5
is an equivalence relation on Z.

Let n be a fixed positive integer. Define a relation R on Z as follows:
(a, b) ∈ R ⇔ a − b is divisible by n.
Show that R is an equivalence relation on Z.

Let Z be the set of integers. Show that the relation
 R = {(a, b) : a, b ∈ Z and a + b is even}
is an equivalence relation on Z.

m is said to be related to n if m and n are integers and m − n is divisible by 13. Does this define an equivalence relation?

Let R be a relation on the set A of ordered pair of integers defined by (x, y) R (u, v) if xv = yu. Show that R is an equivalence relation.

Show that the relation R on the set A = {x ∈ Z ; 0 ≤ x ≤ 12}, given by R = {(a, b) : a = b}, is an equivalence relation. Find the set of all elements related to 1.

Let L be the set of all lines in XY-plane and R be the relation in L defined as R = {L₁, L₂) : L₁ is parallel to L₂}. Show that R is an equivalence relation. Find the set of all lines related to the line y= 2x + 4.

Show that the relation R, defined in the set A of all polygons as R = {(P₁, P₂) : P₁ and P₂ have the same number of sides}, is an equivalence relation. What is the set of all elements in A related to the right-angled triangle T with sides 3, 4 and 5?

Let O be the origin. We define a relation between two points P and Q in a plane if OP = OQ. Show that the relation, so defined is an equivalence relation.

Let R be the relation defined on the set A = {1, 2, 3, 4, 5, 6, 7} by R = {(a, b) : both a and b are either odd or even}. Show that R is an equivalence relation. Further, show that all the elements of the subset {1, 3, 5, 7} are related to each other and all the elements of the subset {2, 4, 6} are related to each other, but no element of the subset {1, 3, 5, 7} is related to any element of the subset {2, 4, 6}.

Let S be a relation on the set R of all real numbers defined by
S = {(a, b) ∈ R × R : a² + b² = 1}
Prove that S is not an equivalence relation on R.

Let Z be the set of all integers and Z₀ be the set of all non-zero integers. Let a relation R on Z × Z₀be defined as (a, b) R (c, d) ⇔ ad = bc for all (a, b), (c, d) ∈ Z × Z₀,
Prove that R is an equivalence relation on Z × Z₀.

If R and S are relations on a set A, then prove that R and S are symmetric ⇒ R ∩ S and R ∪ S are symmetric ?

If R and S are relations on a set A, then prove that R is reflexive and S is any relation ⇒ R ∪ S is reflexive ?

If R and S are transitive relations on a set A, then prove that R ∪ S may not be a transitive relation on A.

Let C be the set of all complex numbers and C₀ be the set of all no-zero complex numbers. Let a relation R on C₀ be defined as

`z_1 R  z_2  ⇔ (z_1 -z_2)/(z_1 + z_2)` is real for all z₁, z₂ ∈ C₀.

Show that R is an equivalence relation.

Write the domain of the relation R defined on the set Z of integers as follows:-
(a, b) ∈ R ⇔ a² + b² = 25

If R = {(x, y) : x² + y² ≤ 4; x, y ∈ Z} is a relation on Z, write the domain of R.

Write the identity relation on set A = {a, b, c}.

Write the smallest reflexive relation on set A = {1, 2, 3, 4}.

If R = {(x, y) : x + 2y = 8} is a relation on N by, then write the range of R.

If R is a symmetric relation on a set A, then write a relation between R and R⁻¹.

Let R = {(x, y) : |x² − y²| <1) be a relation on set A = {1, 2, 3, 4, 5}. Write R as a set of ordered pairs.

If A = {2, 3, 4}, B = {1, 3, 7} and R = {(x, y) : x ∈ A, y ∈ B and x < y} is a relation from A to B, then write R⁻¹.

Let A = {3, 5, 7}, B = {2, 6, 10} and R be a relation from A to B defined by R = {(x, y) : x and y are relatively prime}. Then, write R and R⁻¹.

Define a reflexive relation ?

Define a symmetric relation ?

Define a transitive relation ?

Define an equivalence relation ?

If A = {3, 5, 7} and B = {2, 4, 9} and R is a relation given by "is less than", write R as a set ordered pairs.

A = {1, 2, 3, 4, 5, 6, 7, 8} and if R = {(x, y) : y is one half of x; x, y ∈ A} is a relation on A, then write R as a set of ordered pairs.

Let A = {2, 3, 4, 5} and B = {1, 3, 4}. If R is the relation from A to B given by a R b if "a is a divisor of b". Write R as a set of ordered pairs.

State the reason for the relation R on the set {1, 2, 3} given by R = {(1, 2), (2, 1)} to be transitive ?

Let R = {(a, a³) : a is a prime number less than 5} be a relation. Find the range of R.

Let R be the equivalence relation on the set Z of the integers given by R = { (a, b) : 2 divides a - b }.

Write the equivalence class [0].

For the set A = {1, 2, 3}, define a relation R on the set A as follows:
R = {(1, 1), (2, 2), (3, 3), (1, 3)}
Write the ordered pairs to be added to R to make the smallest equivalence relation.

Let A = {0, 1, 2, 3} and R be a relation on A defined as
R = {(0, 0), (0, 1), (0, 3), (1, 0), (1, 1), (2, 2), (3, 0), (3, 3)}
Is R reflexive? symmetric? transitive?

Let the relation R be defined on the set A = {1, 2, 3, 4, 5} by R = {(a, b) : | a²- b² | < 8}. Write R as a set of ordered pairs.

Let the relation R be defined on N by aRb iff 2a + 3b = 30. Then write R as a set of ordered pairs

Write the smallest equivalence relation on the set A = {1, 2, 3} ?

Let R be a relation on the set N given by
R = {(a, b) : a = b − 2, b > 6}. Then,

If a relation R is defined on the set Z of integers as follows:
(a, b) ∈ R ⇔ a² + b² = 25. Then, domain (R) is ___________

R is a relation on the set Z of integers and it is given by
(x, y) ∈ R ⇔ | x − y | ≤ 1. Then, R is ______________ .

The relation R defined on the set A = {1, 2, 3, 4, 5} by
R = {(a, b) : | a² − b² | < 16} is given by ______________ .

Let R be the relation over the set of all straight lines in a plane such that  l₁ R l₂ ⇔ l ₁⊥ l₂. Then, R is _____________ .

If A = {a, b, c}, then the relation R = {(b, c)} on A is _______________ .

Let A = {2, 3, 4, 5, ..., 17, 18}. Let '≃' be the equivalence relation on A × A, cartesian product of Awith itself, defined by (a, b) ≃ (c, d) if ad = bc. Then, the number of ordered pairs of the equivalence class of (3, 2) is _______________ .

Let A = {1, 2, 3}. Then, the number of relations containing (1, 2) and (1, 3) which are reflexive and symmetric but not transitive is ______.

The relation 'R' in N × N such that
(a, b) R (c, d) ⇔ a + d = b + c is ______________ .

If A = {1, 2, 3}, B = {1, 4, 6, 9} and R is a relation from A to B defined by 'x is greater than y'. The range of R is ______________ .

A relation R is defined from {2, 3, 4, 5} to {3, 6, 7, 10} by : x R y ⇔ x is relatively prime to y. Then, domain of R is ______________ .

A relation ϕ from C to R is defined by x ϕ y ⇔ | x | = y. Which one is correct?

Let R be a relation on N defined by x + 2y = 8. The domain of R is _______________ .

R is a relation from {11, 12, 13} to {8, 10, 12} defined by y = x − 3. Then, R⁻¹ is ______________ .

Let R = {(a, a), (b, b), (c, c), (a, b)} be a relation on set A = a, b, c. Then, R is _______________ .

Let A = {1, 2, 3} and B = {(1, 2), (2, 3), (1, 3)} be a relation on A. Then, R is ________________ .

If R is the largest equivalence relation on a set A and S is any relation on A, then _____________ .

If R is a relation on the set A = {1, 2, 3, 4, 5, 6, 7, 8, 9} given by x R y ⇔ y = 3 x, then R = _____________ .

If R is a relation on the set A = {1, 2, 3} given by R = {(1, 1), (2, 2), (3, 3)}, then R is ____________ .

 If A = {a, b, c, d}, then a relation R = {(a, b), (b, a), (a, a)} on A is _____________ .

If A = {1, 2, 3}, then a relation R = {(2, 3)} on A is _____________ .

Let R be the relation on the set A = {1, 2, 3, 4} given by R = {(1, 2), (2, 2), (1, 1), (4, 4), (1, 3), (3, 3), (3, 2)}. Then, _____________________ .

Let A = {1, 2, 3}. Then, the number of equivalence relations containing (1, 2) is ______.

The relation R = {(1, 1), (2, 2), (3, 3)} on the set {1, 2, 3} is ___________________ .

S is a relation over the set R of all real numbers and it is given by (a, b) ∈ S ⇔ ab ≥ 0. Then, S is _______________ .

In the set Z of all integers, which of the following relation R is not an equivalence relation ?

Mark the correct alternative in the following question:

Let A = {1, 2, 3} and consider the relation R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 3), (1, 3)}. Then, R is _______________ .

Mark the correct alternative in the following question:

The relation S defined on the set R of all real number by the rule aSb if a ≥ b is _______________ .

Mark the correct alternative in the following question:

The maximum number of equivalence relations on the set A = {1, 2, 3} is _______________ .

Mark the correct alternative in the following question:

Let R be a relation on the set N of natural numbers defined by nRm if n divides m. Then, R is _____________ .

Mark the correct alternative in the following question:

Let L denote the set of all straight lines in a plane. Let a relation R be defined by lRm if l is perpendicular to m for all l, m ∈ L. Then, R is ______________ .

Mark the correct alternative in the following question:

Let T be the set of all triangles in the Euclidean plane, and let a relation R on T be defined as aRb if a is congruent to b for all a, b ∈ T. Then, R is ____________ .

Mark the correct alternative in the following question:

Consider a non-empty set consisting of children in a family and a relation R defined as aRb if a is brother of b. Then, R is _____________ .

Mark the correct alternative in the following question:

For real numbers x and y, define xRy if `x-y+sqrt2` is an irrational number. Then the relation R is ___________ .