Advertisements
Advertisements
प्रश्न
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.
Advertisements
उत्तर
R={ (m, n) : m, n ∈ Z, m=kn, where k ∈ N}
Reflexivity :
Let m be an arbitrary elementof R.Then,
m = km is true for k=1
⇒ (m, m) ∈ R
Thus, R is reflexive.
Symmetry: Let (m, n) ∈ R
⇒ m =kn for some k ∈ N
→ `n =1 /km`
⇒ (n, m) ∉ R
Thus, R is not symmetric.
Transitivity : Let (m, n) and (n, o) ∈ R
⇒ m=kn and n=lo for some k, l ∈ N
⇒ m=(kl) o
Here, kl ∈ R
⇒ (m, o) ∈ R
Thus, R is transitive.
APPEARS IN
संबंधित प्रश्न
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}
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, symmetric and transitive ?
Is it true that every relation which is symmetric and transitive is also reflexive? Give reasons.
Give an example of a relation which is reflexive and symmetric but not transitive?
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 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.
If R = {(x, y) : x + 2y = 8} is a relation on N by, then write the range of R.
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 symmetric 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.
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.
If A = {a, b, c}, then the relation R = {(b, c)} on A is _______________ .
If R is the largest equivalence relation on a set A and S is any relation on A, then _____________ .
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 _______________ .
For the matrix A = `[(2,3),(5,7)]`, find (A + A') and verify that it is a symmetric matrix.
If A = {1, 2, 3, 4 }, define relations on A which have properties of being:
reflexive, transitive but not symmetric
Let R be relation defined on the set of natural number N as follows:
R = {(x, y): x ∈N, y ∈N, 2x + y = 41}. Find the domain and range of the relation R. Also verify whether R is reflexive, symmetric and transitive
Give an example of a map which is neither one-one nor onto
The following defines a relation on N:
x + y = 10, x, y ∈ N
Determine which of the above relations are reflexive, symmetric and transitive.
The maximum number of equivalence relations on the set A = {1, 2, 3} are ______.
Let the relation R be defined on the set A = {1, 2, 3, 4, 5} by R = {(a, b) : |a2 – b2| < 8. Then R is given by ______.
Let `"f"("x") = ("x" - 1)/("x" + 1),` then f(f(x)) is ____________.
Let us define a relation R in R as aRb if a ≥ b. Then R is ____________.
Let S = {1, 2, 3, 4, 5} and let A = S x S. Define the relation R on A as follows:
(a, b) R (c, d) iff ad = cb. Then, R is ____________.
Total number of equivalence relations defined in the set S = {a, b, c} is ____________.
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}
- Raji wants to know the number of relations possible from A to B. How many numbers of relations are possible?
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: B → B be defined by R = {(1,1),(1,2), (2,2), (3,3), (4,4), (5,5), (6,6)}, then R is ____________.
Students of Grade 9, planned to plant saplings along straight lines, parallel to each other to one side of the playground ensuring that they had enough play area. Let us assume that they planted one of the rows of the saplings along the line y = x − 4. Let L be the set of all lines which are parallel on the ground and R be a relation on L.
Answer the following using the above information.
- Let R = `{ ("L"_1, "L"_2) ∶ "L"_1 bot "L"_2 "where" "L"_1, "L"_2 in "L" }` which of the following is true?
The relation R = {(1,1),(2,2),(3,3)} on {1,2,3} is ____________.
Find: `int (x + 1)/((x^2 + 1)x) dx`
Which one of the following relations on the set of real numbers R is an equivalence relation?
On the set N of all natural numbers, define the relation R by a R b, if GCD of a and b is 2. Then, R is
A relation 'R' in a set 'A' is called reflexive, if
Given a non-empty set X, define the relation R in P(X) as follows:
For A, B ∈ P(X), (4, B) ∈ R iff A ⊂ B. Prove that R is reflexive, transitive and not symmetric.
