Advertisements
Advertisements
प्रश्न
Three relations R1, R2 and R3 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, c)}
R4 = {(a, b), (b, c), (c, a)}.
Find whether or not each of the relations R1, R2, R3, R4 on A is (i) reflexive (ii) symmetric and (iii) transitive.
उत्तर
(i) R1
Reflexive:
Clearly, (a, a), (b, b) and (c, c) ∈ R1
So, R1 is reflexive.
Symmetric:
We see that the ordered pairs obtained by interchanging the components of R1 are also in R1.
So, R1 is not symmetric.
Transitive:
Here,
(a, b)∈R1, (b, c)∈R1 and also (a, c)∈R1
So, R1 is transitive.
(ii) R2
Reflexive: Clearly (a,a) ∈R2 . So, R2 is reflexive.
Symmetric: Clearly (a,a) ∈R ⇒ (a,a) ∈R. So, R2 is symmetric.
Transitive: R2 is clearly a transitive relation, since there is only one element in it.
iii) R3
Reflexive:
Here,
(b, b)∉ R3 neither (c, c) ∉ R3
So, R3 is not reflexive.
Symmetric:
Here,
(b, c) ∈R3, but (c,b) ∉R3
So,R3isnotsymmetric.
Transitive:
Here, R3 has only two elements. Hence, R3 is transitive.
(iv) R4
Reflexive:
Here,
(a, a) ∉ R4, (b, b)∉ R4 (c, c)∉ R4
So, R4 is not reflexive.
Symmetric:
Here,
(a, b)∈ R4, but (b,a) ∉ R4.
So, R4 is not symmetric
Transitive:
Here,
(a, b)∈R4, (b, c)∈R4, but (a, c)∉R4
So, R4 is not transitive.
APPEARS IN
संबंधित प्रश्न
determination of whether the following relations are reflexive, symmetric, and transitive:
Relation R in the set A = {1, 2, 3...13, 14} defined as R = {(x,y):3x - y = 0}
Show that the relation R in the set {1, 2, 3} given by R = {(1, 2), (2, 1)} is symmetric but neither reflexive nor 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.
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 triangles as R = {(T1, T2): T1 is similar to T2}, is equivalence relation. Consider three right angle triangles T1 with sides 3, 4, 5, T2 with sides 5, 12, 13 and T3 with sides 6, 8, and 10. Which triangles among T1, T2 and T3 are related?
Let R be the relation in the set {1, 2, 3, 4} given by R = {(1, 2), (2, 2), (1, 1), (4, 4), (1, 3), (3, 3), (3, 2)}. Choose the correct answer.
Test whether the following relation R3 is (i) reflexive (ii) symmetric and (iii) transitive:
R3 on R is defined by (a, b) ∈ R3 `⇔` a2 – 4ab + 3b2 = 0.
The following relation is defined on the set of real numbers. aRb if |a| ≤ b
Find whether relation is reflexive, symmetric or 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.
Give an example of a relation which is symmetric and transitive but not reflexive?
Give an example of a relation which is symmetric but neither reflexive nor transitive?
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 it reflexive and transitive.
Defines a relation on N :
x + y = 10, x, y∈ N
Determine the above relation is reflexive, symmetric and transitive.
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.
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.
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.
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.
Mark the correct alternative in the following question:
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 for all a, b T. Then, R is ____________ .
Mark the correct alternative in the following question:
Consider a non-empty set consisting of children in a family and a relation R defined as aRb if a is brother of b. Then, R is _____________ .
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 :
R1 = {(a, a2) / a is prime number less than 15}
Let R be a relation on the set N of natural numbers defined by nRm if n divides m. Then R is ______.
If A = {1, 2, 3, 4 }, define relations on A which have properties of being:
reflexive, transitive but not symmetric
Give an example of a map which is not one-one but onto
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 ______.
Which of the following is not an equivalence relation on I, the set of integers: x, y
Let A = {1, 2, 3, 4, 5, 6} Which of the following partitions of A correspond to an equivalence relation on A?
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: B → B be defined by R = {(1,1),(1,2), (2,2), (3,3), (4,4), (5,5), (6,6)}, then R is ____________.
An organization conducted a bike race under 2 different categories-boys and girls. Totally there were 250 participants. Among all of them finally, three from Category 1 and two from Category 2 were selected for the final race. Ravi forms two sets B and G with these participants for his college project. Let B = {b1,b2,b3} G={g1,g2} where B represents the set of boys selected and G the set of girls who were selected for the final race.
Ravi decides to explore these sets for various types of relations and functions.
- Ravi wishes to form all the relations possible from B to G. How many such relations are possible?
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 in a set 'A' is known as empty relation:-
A relation 'R' in a set 'A' is called a universal relation, if each element of' A' is related to :-
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.
If a relation R on the set {a, b, c} defined by R = {(b, b)}, then classify the relation.
