Advertisements
Advertisements
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)].
Advertisements
उत्तर
A = {1, 2, 3, ..., 9} ⊂ ℕ, the set of natural numbers
Let 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.
We have to show that R is an equivalence relation.
Reflexivity:
Let (a, b) be an arbitrary element of A × A. Then, we have:
(a, b) ∈ A × A
⇒ a, b ∈ A
⇒ a + b = b + a (by commutativity of addition on A ⊂ ℕ)
⇒ (a, b) R (a, b)
Thus, (a, b) R (a, b) for all (a, b) ∈ A × A.
So, R is reflexive.
Symmetry:
Let (a, b), (c, d) ∈ A × A such that (a, b) R (c, d).
a + d = b + c
⇒ b + c = a + d
⇒ c + b = d + a (by commutativity of addition on A ⊂ ℕ)
⇒ (c, d) R (a, b)
Thus, (a, b) R (c, d) ⇒ (c, d) R (a, b) for all (a, b), (c, d) ∈ A × A.
So, R is symmetric
Transitivity:
Let (a, b), (c, d), (e, f) ∈ A × A such that (a, b) R (c, d) and (c, d) R (e, f). Then, we have:
(a, b) R (c, d)
⇒ a + d = b + c ... (1)
(c, d) R (e, f)
⇒ c + f = d + e ... (2)
Adding equations (1) and (2), we get:
(a + d) + (c + f) = (b + c) + (d + e)
⇒ a + f = b + e
⇒ (a, b) R (e, f)
Thus, (a, b) R (c, d) and (c, d) R (e, f) ⇒ (a, b) R (e, f) for all (a, b), (c, d), (e, f) ∈ A × A.
So, R is transitive on A × A.
Thus, R is reflexive, symmetric and transitive.
∴ R is an equivalence relation.
To write the equivalence class of [(2, 5)], we need to search all the elements of the type (a, b) such that 2 + b = 5 + a.
∴ Equivalence class of [(2, 5)] = {(1, 4), (2, 5), (3, 6), (4, 7), (5, 8), (6, 9)}
APPEARS IN
संबंधित प्रश्न
Determine whether the following relation is reflexive, symmetric and transitive:
Relation R in the set N of natural numbers defined as R = {(x, y) : y = x + 5 and x < 4}.
Let A = {1, 2, 3}. Then number of equivalence relations containing (1, 2) is ______.
Test whether the following relation R1 is (i) reflexive (ii) symmetric and (iii) transitive :
R1 on Q0 defined by (a, b) ∈ R1 ⇔ a = 1/b.
If A = {1, 2, 3, 4} define relations on A which have properties of being symmetric but neither reflexive nor transitive ?
m is said to be related to n if m and n are integers and m − n is divisible by 13. Does this define an equivalence relation?
Show that the relation R on the set A = {x ∈ Z ; 0 ≤ x ≤ 12}, given by R = {(a, b) : a = b}, is an equivalence relation. Find the set of all elements related to 1.
If A = {2, 3, 4}, B = {1, 3, 7} and R = {(x, y) : x ∈ A, y ∈ B and x < y} is a relation from A to B, then write R−1.
Define a reflexive relation ?
Define an equivalence relation ?
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 ______________ .
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 on R defined as R = {(a, b): a ≤ b}, is reflexive, and transitive but not symmetric.
If A = {a, b, c}, B = (x , y} find B × A.
Write the relation in the Roster form and hence find its domain and range :
R1 = {(a, a2) / a is prime number less than 15}
If A = {1, 2, 3, 4 }, define relations on A which have properties of being:
reflexive, symmetric and transitive
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
Let us define a relation R in R as aRb if a ≥ b. Then R is ______.
Every relation which is symmetric and transitive is also reflexive.
Which of the following is not an equivalence relation on I, the set of integers: x, y
If f(x) = `1 - 1/"x", "then f"("f"(1/"x"))` ____________.
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 ____________.
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 ____________.
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
If f(x + 2a) = f(x – 2a), then f(x) is:
Which relation is reflexive on \(A=\{1,2\}\)?
A relation \(R\) is transitive if which condition is satisfied?
Which pair of relations are called trivial relations?
Which statement about the Identity Relation is correct?
