Advertisements
Advertisements
प्रश्न
Give an example of a relation which is symmetric and transitive but not reflexive?
Advertisements
उत्तर
Let A = {−5, −6}
The relation R on a set A is defined as follows:
R = {−5, −6), (−6, −5), (−5, −5)}
The relation R is not reflexive because (−6, −6) ∉ R.
∴ R is not reflexive.
⇒ The relation R is symmetric because (−5, −6) ∈ R and (−6, −5) ∈ R.
∴ R is symmetric.
⇒ And, if (−5, −6) and (−6, −5) ∈ R, then (−5, −5) ∈ R
∴ R is transitive.
Hence, the relation R is symmetric and transitive but not reflexive.
APPEARS IN
संबंधित प्रश्न
Let A = {1, 2, 3,......, 9} and R be the relation in A × A defined by (a, b) R (c, d) if a + d = b + c for (a, b), (c, d) in A × A. Prove that R is an equivalence relation. Also, obtain the equivalence class [(2, 5)].
Show that the relation R in the set R of real numbers, defined as R = {(a, b) : a ≤ b2} is neither reflexive nor symmetric nor transitive.
Check whether the relation R in R defined by R = {(a, b) : a ≤ b3} is reflexive, symmetric or transitive.
Show that the relation R defined in the set A of all triangles as R = {(T1, T2) : T1 is similar to T2}, is an 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?
If A = {1, 2, 3, 4} define relations on A which have properties of being symmetric but neither reflexive nor 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 transitive but neither reflexive nor symmetric?
Let S be a relation on the set R of all real numbers defined by
S = {(a, b) ∈ R × R : a2 + b2 = 1}
Prove that S is not an equivalence relation on R.
Let Z be the set of all integers and Z0 be the set of all non-zero integers. Let a relation R on Z × Z0be defined as (a, b) R (c, d) ⇔ ad = bc for all (a, b), (c, d) ∈ Z × Z0,
Prove that R is an equivalence relation on Z × Z0.
If R and S are relations on a set A, then prove that R and S are symmetric ⇒ R ∩ S and R ∪ S are symmetric ?
Write the smallest reflexive relation on set A = {1, 2, 3, 4}.
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.
R is a relation from {11, 12, 13} to {8, 10, 12} defined by y = x − 3. Then, R−1 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 maximum number of equivalence relations on the set A = {1, 2, 3} is _______________ .
Show that the relation R defined by (a, b)R(c,d) ⇒ a + d = b + c on the A x A , where A = {1, 2,3,...,10} is an equivalence relation. Hence write the equivalence class [(3, 4)]; a, b, c,d ∈ A.
Let A = {1, 2, 3, 4}, B = {4, 5, 6}, C = {5, 6}. Find A × (B ∪ C).
Let A = {6, 8} and B = {1, 3, 5}.
Let R = {(a, b)/a∈ A, b∈ B, a – b is an even number}. Show that R is an empty relation from A to B.
Consider the set A = {1, 2, 3} and R be the smallest equivalence relation on A, then R = ______
Consider the set A = {1, 2, 3} and the relation R = {(1, 2), (1, 3)}. R is a transitive relation.
Let n be a fixed positive integer. Define a relation R in Z as follows: ∀ a, b ∈ Z, aRb if and only if a – b is divisible by n. Show that R is an equivalance relation
If A = {1, 2, 3, 4 }, define relations on A which have properties of being:
symmetric but neither reflexive nor transitive
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 neither one-one nor onto
The following defines a relation on N:
x is greater than y, x, y ∈ N
Determine which of the above relations are reflexive, symmetric and transitive.
Let A = { 2, 3, 6 } Which of the following relations on A are reflexive?
If f(x) = `1 - 1/"x", "then f"("f"(1/"x"))` ____________.
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 ____________.
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
There are 600 student in a school. If 400 of them can speak Telugu, 300 can speak Hindi, then the number of students who can speak both Telugu and Hindi is:
Let A = {1, 2, 3, 4} and let R = {(2, 2), (3, 3), (4, 4), (1, 2)} be a relation on A. Then R is ______.
If a relation R on the set {a, b, c} defined by R = {(b, b)}, then classify the relation.
Which statement defines a Universal Relation?
Which pair of relations are called trivial relations?
