Advertisements
Advertisements
Question
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)].
Solution
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
RELATED QUESTIONS
Given an example of a relation. Which is Transitive but neither reflexive nor symmetric.
Given an example of a relation. Which is Reflexive and transitive but not symmetric.
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?
The binary operation *: R x R → R is defined as a *b = 2a + b Find (2 * 3)*4
Defines a relation on N :
xy is square of an integer, 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 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 transitive relations on a set A, then prove that R ∪ S may not be a transitive relation on A.
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 ?
For the set A = {1, 2, 3}, define a relation R on the set A as follows:
R = {(1, 1), (2, 2), (3, 3), (1, 3)}
Write the ordered pairs to be added to R to make the smallest equivalence relation.
Write the smallest equivalence relation on the set A = {1, 2, 3} ?
Let R be a relation on N defined by x + 2y = 8. The domain of R is _______________ .
Consider the set A = {1, 2, 3} and R be the smallest equivalence relation on A, then R = ______
Let A = { 2, 3, 6 } Which of the following relations on A are reflexive?
Which of the following is not an equivalence relation on I, the set of integers: x, y
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 ____________.
Total number of equivalence relations defined in the set S = {a, b, c} is ____________.
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?
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:
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}
- Raji wants to know the number of relations possible from A to B. How many numbers of relations are possible?
Students of Grade 9, planned to plant saplings along straight lines, parallel to each other to one side of the playground ensuring that they had enough play area. Let us assume that they planted one of the rows of the saplings along the line y = x − 4. Let L be the set of all lines which are parallel on the ground and R be a relation on L.
Answer the following using the above information.
- Let R = `{ ("L"_1, "L"_2) ∶ "L"_1 bot "L"_2 "where" "L"_1, "L"_2 in "L" }` which of the following is true?
The relation > (greater than) on the set of real numbers is
A relation in a set 'A' is known as empty relation:-
Let R = {(x, y) : x, y ∈ N and x2 – 4xy + 3y2 = 0}, where N is the set of all natural numbers. Then the relation R is ______.
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 ______.
Let N be the set of all natural numbers and R be a relation on N × N defined by (a, b) R (c, d) `⇔` ad = bc for all (a, b), (c, d) ∈ N × N. Show that R is an equivalence relation on N × N. Also, find the equivalence class of (2, 6), i.e., [(2, 6)].
If a relation R on the set {a, b, c} defined by R = {(b, b)}, then classify the relation.
