Advertisements
Advertisements
प्रश्न
Is it true that every relation which is symmetric and transitive is also reflexive? Give reasons.
उत्तर
No, it is not true.
Consider a set A = {1, 2, 3, 4} and define a relation R on A.
Symmetric relation:
R = {(1, 2), (2, 1)} is symmetric on set A.
Transitive relation:
R = {(1, 2), (2, 1), (1, 1)} is the simplest transitive relation on set A.
R = {(1, 2), (2, 1), (1, 1)} is symmetric as well as transitive relation.
But R is not reflexive here.
If only (2, 2) ∈ R, had it been reflexive.
Thus, it is not true that every relation which is symmetric and transitive is also reflexive.
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 of points in a plane given by R = {(P, Q): distance of the point P from the origin is same as the distance of the point Q from the origin}, is an equivalence relation. Further, show that the set of all points related to a point P ≠ (0, 0) is the circle passing through P with origin as centre.
Show that the relation R defined in the set A of all polygons as R = {(P1, P2): P1 and P2have 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?
Given a non-empty set X, consider P (X), which is the set of all subsets of X. Define the relation R in P(X) as follows:
For subsets A, B in P(X), ARB if and only if A ⊂ B. Is R an equivalence relation on P(X)? Justify your answer.
The binary operation *: R x R → R is defined as a *b = 2a + b Find (2 * 3)*4
The following relation is defined on the set of real numbers.
aRb if 1 + ab > 0
Find whether relation is reflexive, symmetric or transitive.
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.
Give an example of a relation which is reflexive and symmetric but not transitive ?
Give an example of a relation which is symmetric but neither reflexive nor transitive?
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.
Defines a relation on N :
x + y = 10, 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 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 = {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 N by aRb iff 2a + 3b = 30. Then write R as a set of ordered pairs
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 _____________ .
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 ______.
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 ______________ .
If A = {1, 2, 3}, then a relation R = {(2, 3)} on A is _____________ .
Let A = {1, 2, 3}. Then, the number of equivalence relations containing (1, 2) is ______.
Show that the relation R on the set Z of all integers, given by R = {(a,b) : 2 divides (a-b)} is an equivalence relation.
If A = {a, b, c}, B = (x , y} find A × A.
Let A = {1, 2, 3, 4}, B = {4, 5, 6}, C = {5, 6} Find (A × B) ∩ (A × C).
Let A = {1, 2, 3}. Which of the following is not an equivalence relation on A?
Let A = {1, 2, 3}, then the relation R = {(1, 1), (1, 2), (2, 1)} on A is ____________.
If A is a finite set containing n distinct elements, then the number of relations on A is equal to ____________.
Let us define a relation R in R as aRb if a ≥ b. Then R is ____________.
The relation R is defined on the set of natural numbers as {(a, b) : a = 2b}. Then, R-1 is given by ____________.
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:
Which one of the following relations on the set of real numbers R is an equivalence relation?
In a group of 52 persons, 16 drink tea but not coffee, while 33 drink tea. How many persons drink coffee but not tea?
A market research group conducted a survey of 2000 consumers and reported that 1720 consumers like product P1 and 1450 consumers like product P2. What is the least number that must have liked both the products?
If f(x + 2a) = f(x – 2a), then f(x) is:
Let A = {3, 5}. Then number of reflexive relations on A is ______.
Let L be a set of all straight lines in a plane. The relation R on L defined as 'perpendicular to' is ______.
