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Let A be the set of all human beings in a town at a particular time. Determine whether each of the following relations are reflexive, symmetric and transitive:
R = {(x, y) : x and y work at the same place}
R = {(x, y) : x and y live in the same locality}
R = {(x, y) : x is wife of y}
R = {(x, y) : x is father of y}
Relations R1, R2, R3 and R4 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, a)}
R4 = {(a, b) (b, c) (c, a)}
Find whether or not each of the relations R1, R2, R3, R4 on A is (i) reflexive (iii) symmetric (iii) transitive.
Test whether the following relations R1, R2 and R3 are (i) reflexive (ii) symmetric and (iii) transitive :
R1 on Q0 defined by (a, b) ϵ R1⇔ a = 1/b
R2 on Z defined by (a, b) ϵ R2⇔ |a – b| ≤ 5
R3 on R defined by (a, b) ϵ R3⇔ a2 – 4 ab + 3b2 = 0.
Let A = {1, 2, 3}, and let R1 = {(1, 1), (1, 3), (3, 1), (2, 2), (2, 1), (3, 3), (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 relations are defined on the set of real numbers :
aRb if a – b > 0
Find whether these relations are reflexive, symmetric or transitive.
aRb if 1 + ab > 0
aRb if | a | ≤ b.
Check whether the relation R defined on the set A={1,2,3,4,5,6} as R= {(a, b) : b = a + 1} is reflexive, symmetric or transitive.
Check whether the relation R on 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.
symmetric but neither reflexive nor transitive.
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.
reflexive and transitive but not symmetric.
symmetric and transitive but not reflexive.
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 ordered pairs so that the enlarged relation is symmetric, 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 a minimum number of ordered pairs to be added in R to make it reflexive and transitive.
Each of the following defines a relation on N :
x > y, x, y ∈ N
Determine which of the above relations are reflexive, symmetric and transitive.
x + y = 10, x, y ∈ N
xy is square of an integer, x, y ∈ N
x + 4y = 10, x, y ∈ N
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 pairs of non-zero integers defined by (x, y) R (u, v) iff 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 on the set A of all polygons as
R = {(P1, P2) : P1 and P2 have same number of sides}, is an equivalence relation. What is the set of all elements in A related to the right angle 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 × Z0 be defined as follows :
(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 the following :
(i) R and S are symmetric ⇔ R ⋂ S, and R ⋃ S is symmetric
(ii) 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 non-zero complex numbers. Let a relation R on C0 be defined as
z1 R z2 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) ϵR ⬄ a2 + b2 = 25
If R = {(x,y): x2 + y2 ≤ 4; x,y ϵ Z} is a relation of Z, write the domain of R.
Write the identity relation of set A = {a, b, c}.
Write the smallest reflexive relation of set A = {1, 2, 3, 4}.
If R = {(x,y) : x + 2y = 8} is a relation on N, 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 ϵ A, y ϵB 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 ϵA} 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 iff “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)} not to be transitive.
Let R = {(a, a3) : a is a prime number less than 5} be a relation. Find the range of R.
Show that R is an equivalence relation on the set Z of 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
F = {(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 a Rb 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
Which of the following is not an equivalence relation on Z?
R is a relation on the set Z of integers and it is given by (x, y) ϵ R ⬄ |x – y| ≤ 1. Then, R is
The relation R defined on the set A = {1, 2, 3, 4, 5} by R = is given by
Let R be the relation over the set of all straight lines in a plane such that ℓ1 R ℓ2⬄ ℓ1⊥ 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 A with itself, defined by (a, b) ≃ (c, d) iff 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, 5, 9} and R is a relation from A to B defined by ‘x is greater than y’. Then, Range of R is
A relation R is defined form {2, 3, 4, 5} to {3, 6, 7, 10} by xRy ⬄ x is relatively prime to y. Then, domain of R is
A relation ϕ from C to R is defined by x ϕ ⬄ |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 R = {(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 = 3x, 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 its is given by (a, b) ϵ S ⬄ ab ≥ 0. Then, S is
If the set Z of all integers, which of the following relation R is not an equivalence relation?
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
The relation S defined on the set R of all real number by the rule a Sb iff a ≥ b is
The maximum number of equivalence relations on the set A = {1, 2, 3} is
Let R be a relation on the set N of natural numbers defined by n R m iff n divides m. Then, R is
Let L denote the set of all straight lines in a plane. Let a relation R be defined by ℓ R m iff ℓ is perpendicular to m for all ℓ, m ϵ L. Then, R is
Let T be the set of all triangles in the Euclidean plane, and let a relation R on T be defined as a R b if a is congruent to b for all a, b ϵ T. Then, R is
Consider a non-empty set consisting of children in a family and a relation R defined as a R B if a is brother of b. Then, R is
For real numbers x and y, define x R y iff x – y+ √2 is an irrational number. Then the relation R is
