Advertisements
Advertisements
Questions
Let A be the set of all human beings in a town at a particular time. Determine whether of the following relation is reflexive, symmetric and transitive :
R = {(x, y) : x and y live in the same locality}
determination of whether the following relations are reflexive, symmetric, and transitive:
Relation R in the set A of human beings in a town at a particular time given by R = {(x, y) : x and y live in the same locality}
Solution
(i) Reflexivity:
Let x be an arbitrary element of R. Then,
x ∈ R
⇒ x and x live in the same locality is true since they are the same.
So, R is a reflexive relation.
(ii) Symmetry:
Let (x, y) ∈ R
⇒ x and y live in the same locality
⇒ y and x live in the same locality
⇒ (y, x) ∈ R
So, R is a symmetric relation.
(iii) Transitivity:
Let (x, y)∈R and (y, z)∈R. Then,
x and y live in the same locality and y and z live in the same locality
⇒ x, y and z all live in the same locality
⇒ x and z live in the same locality
⇒ (x, z) ∈ R
So, R is a transitive relation.
APPEARS IN
RELATED QUESTIONS
determination of whether the following relations are reflexive, symmetric, and transitive:
Relation R in the set A of human beings in a town at a particular time given by (c) R = {(x, y): x is exactly 7 cm taller than y}
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.
Show that the relation R in the set A of all the books in a library of a college, given by R = {(x, y): x and y have same number of pages} is an equivalence relation.
Given an example of a relation. Which is Reflexive and transitive but not symmetric.
Let A = {1, 2, 3}. Then number of relations containing (1, 2) and (1, 3) which are reflexive and symmetric but not transitive is
(A) 1 (B) 2 (C) 3 (D) 4
The binary operation *: R x R → R is defined as a *b = 2a + b Find (2 * 3)*4
If A = {1, 2, 3, 4} define relations on A which have properties of being reflexive, symmetric and transitive ?
Show that the relation '≥' on the set R of all real numbers is reflexive and transitive but not symmetric ?
Defines a relation on N :
xy is square of an integer, x, y ∈ N
Determine the above relation is reflexive, symmetric and transitive.
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.
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.
Define a reflexive relation ?
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.
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 ______.
R is a relation from {11, 12, 13} to {8, 10, 12} defined by y = x − 3. Then, R−1 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} given by R = {(1, 1), (2, 2), (3, 3)}, then R 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:
The relation S defined on the set R of all real number by the rule aSb if a b is _______________ .
Show that the relation R on the set Z of integers, given by R = {(a,b):2divides (a - b)} is an equivalence relation.
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) ∪ (A × C).
Write the relation in the Roster form and hence find its domain and range :
R1 = {(a, a2) / a is prime number less than 15}
In the set of natural numbers N, define a relation R as follows: ∀ n, m ∈ N, nRm if on division by 5 each of the integers n and m leaves the remainder less than 5, i.e. one of the numbers 0, 1, 2, 3 and 4. Show that R is equivalence relation. Also, obtain the pairwise disjoint subsets determined by R
Let L denote the set of all straight lines in a plane. Let a relation R be defined by lRm if and only if l is perpendicular to m ∀ l, m ∈ L. Then R is ______.
If A = {1, 2, 3, 4 }, define relations on A which have properties of being:
reflexive, symmetric and transitive
Give an example of a map which is one-one but not onto
The following defines a relation on N:
x + 4y = 10 x, y ∈ N.
Determine which of the above relations are reflexive, symmetric and transitive.
Every relation which is symmetric and transitive is also reflexive.
Let A = { 2, 3, 6 } Which of the following relations on A are reflexive?
R = {(1, 1), (2, 2), (1, 2), (2, 1), (2, 3)} be a relation on A, 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 aRb if a is congruent to b ∀ a, b ∈ T. Then R is ____________.
Let R be a relation on the set N of natural numbers denoted by nRm ⇔ n is a factor of m (i.e. n | m). Then, R is ____________.
Given set A = {a, b, c}. An identity relation in set A 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}
- 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 ____________.
The value of k for which the system of equations x + ky + 3z = 0, 4x + 3y + kz = 0, 2x + y + 2z = 0 has nontrivial solution is
Let R = {(a, b): a = a2} for all, a, b ∈ N, then R salifies.
Let f(x)= ax2 + bx + c be such that f(1) = 3, f(–2) = λ and f(3) = 4. If f(0) + f(1) + f(–2) + f(3) = 14, then λ is equal to ______.
lf A = {x ∈ z+ : x < 10 and x is a multiple of 3 or 4}, where z+ is the set of positive integers, then the total number of symmetric relations on A is ______.
