Advertisements
Advertisements
प्रश्न
If R is a relation on the set A = {1, 2, 3} given by R = {(1, 1), (2, 2), (3, 3)}, then R is ____________ .
पर्याय
reflexive
symmetric
transitive
all the three options
Advertisements
उत्तर
all the three options
R={ (a, b) : a=b and a, b ∈ A }
Reflexivity: Let a ∈ A. Then,
a = a
⇒ (a, a) ∈ R for all a ∈ A
So, R is reflexive on A.
Symmetry : Let a, b ∈ A such that (a, b) ∈ R. Then,
(a, b) ∈ R
⇒ a = b
⇒ b = a
⇒ (b, a)∈ R for all a ∈ A
So, R is symmetric on A.
Transitivity : Let a, b, c ∈ A such that (a, b) ∈ R and (b, c) ∈ R. Then,
(a, b) ∈ R ⇒ a = b
and (b, c) ∈ R ⇒ b = c
⇒ a = c
⇒ (a, c)∈ R for all a ∈ A
So, R is transitive on A.
Hence, R is an equivalence relation on A.
APPEARS IN
संबंधित प्रश्न
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}.
Show that the relation R in the set A = {x ∈ Z : 0 ≤ x ≤ 12} given by R = {(a, b) : |a – b| is a multiple of 4} is an equivalence relation. Find the set of all elements related to 1.
Given an example of a relation. Which is symmetric but neither reflexive nor transitive.
If A = {1, 2, 3, 4} define relations on A which have properties of being reflexive, transitive but not symmetric ?
If R is a symmetric relation on a set A, then write a relation between R and R−1.
Define a transitive relation ?
Define an equivalence relation ?
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 R = {(a, a3) : a is a prime number less than 5} be a relation. Find the range of R.
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.
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,
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 _______________ .
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:
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 ______________ .
If A = {a, b, c}, B = (x , y} find A × B.
Let A = {1, 2, 3, 4}, B = {4, 5, 6}, C = {5, 6}. Find A × (B ∪ C).
Write the relation in the Roster form and hence find its domain and range:
R2 = `{("a", 1/"a") "/" 0 < "a" ≤ 5, "a" ∈ "N"}`
Consider the set A = {1, 2, 3} and the relation R = {(1, 2), (1, 3)}. R is a transitive relation.
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 R reflexive and transitive
Given A = {2, 3, 4}, B = {2, 5, 6, 7}. Construct an example of the following:
a mapping from A to B which is not injective
Let us define a relation R in R as aRb if a ≥ b. Then R is ______.
Let R = {(3, 1), (1, 3), (3, 3)} be a relation defined on the set A = {1, 2, 3}. Then R is symmetric, transitive but not reflexive.
Every relation which is symmetric and transitive is also reflexive.
Let A = {1, 2, 3, …. n} and B = {a, b}. Then the number of surjections from A into B is ____________.
Total number of equivalence relations defined in the set S = {a, b, c} is ____________.
A relation R in set A = {1, 2, 3} is defined as R = {(1, 1), (1, 2), (2, 2), (3, 3)}. Which of the following ordered pair in R shall be removed to make it an equivalence relation in A?
Given set A = {1, 2, 3} and a relation R = {(1, 2), (2, 1)}, the relation R will be ____________.
Sherlin and Danju are playing Ludo at home during Covid-19. While rolling the dice, Sherlin’s sister Raji observed and noted the possible outcomes of the throw every time belongs to set {1,2,3,4,5,6}. Let A be the set of players while B be the set of all possible outcomes.
A = {S, D}, B = {1,2,3,4,5,6}
- Let R be a relation on B defined by R = {(1,2), (2,2), (1,3), (3,4), (3,1), (4,3), (5,5)}. Then R is:
Sherlin and Danju are playing Ludo at home during Covid-19. While rolling the dice, Sherlin’s sister Raji observed and noted the possible outcomes of the throw every time belongs to set {1,2,3,4,5,6}. Let A be the set of players while B be the set of all possible outcomes.
A = {S, D}, B = {1,2,3,4,5,6}
- Raji wants to know the number of relations possible from A to B. How many numbers of relations are possible?
If A is a finite set consisting of n elements, then the number of reflexive relations on A is
In a group of 52 persons, 16 drink tea but not coffee, while 33 drink tea. How many persons drink coffee but not tea?
Let N be the set of all natural numbers and R be a relation on N × N defined by (a, b) R (c, d) `⇔` ad = bc for all (a, b), (c, d) ∈ N × N. Show that R is an equivalence relation on N × N. Also, find the equivalence class of (2, 6), i.e., [(2, 6)].
Which statement defines a Universal Relation?
A relation \(R\) is symmetric if which condition is satisfied?
A relation \(R\) is transitive if which condition is satisfied?
