Advertisements
Advertisements
Question
Show that the relation R in 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.
Advertisements
Solution
R = {(a, b) : a = b}
(i) Reflexive:
R = {(0, 0), (1, 1), ..., (12, 12)}
A = {0, 1, 2, ..., 12}
For any element a ∈ A, we have (a, a) ∈ R, since a = a.
∴ R is reflexive.
(ii) Symmetric:
Now, let (a, b) ∈ R.
⇒ a = b
⇒ b = a
⇒ (b, a) ∈ R
∴ R is symmetric.
(iii) Transitive:
Now, let (a, b) ∈ R and (b, c) ∈ R.
⇒ a = b and b = c
⇒ a = c
⇒ (a, c) ∈ R
∴ R is transitive.
Hence, R is an equivalence relation.
The elements in R that are related to 1 will be those elements from set A which are equal to 1.
Hence, the set of elements related to 1 is {1}.
APPEARS IN
RELATED QUESTIONS
Determine whether the following relation is reflexive, symmetric and transitive:
Relation R in the set A = {1, 2, 3, ..., 13, 14} defined as R = {(x, y) : 3x − y = 0}.
Determine whether the following relation is reflexive, symmetric and transitive:
Relation R in the set Z of all integers defined as R = {(x, y) : x − y is an integer}.
Check whether the relation R in R defined by R = {(a, b) : a ≤ b3} is reflexive, symmetric or transitive.
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}.
Given an example of a relation. Which is Symmetric but neither reflexive nor 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?
Test whether the following relation R2 is (i) reflexive (ii) symmetric and (iii) transitive:
R2 on Z defined by (a, b) ∈ R2 ⇔ |a – b| ≤ 5
The following relation is defined on the set of real numbers. aRb if |a| ≤ b
Find whether relation is reflexive, symmetric or transitive.
Give an example of a relation which is transitive but neither reflexive nor symmetric?
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.
Show that the relation R, defined in the set A of all polygons as R = {(P1, P2) : P1 and P2 have the same number of sides}, is an equivalence relation. What is the set of all elements in A related to the right-angled triangle T with sides 3, 4 and 5?
Write the smallest reflexive relation on set A = {1, 2, 3, 4}.
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 symmetric relation ?
If A = {3, 5, 7} and B = {2, 4, 9} and R is a relation given by "is less than", write R as a set ordered pairs.
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.
State the reason for the relation R on the set {1, 2, 3} given by R = {(1, 2), (2, 1)} to be transitive ?
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.
If A = {1, 2, 3}, B = {1, 4, 6, 9} and R is a relation from A to B defined by 'x is greater than y'. The range of R is ______________ .
Let R = {(a, a), (b, b), (c, c), (a, b)} be a relation on set A = a, b, c. Then, R is _______________ .
Mark the correct alternative in the following question:
The maximum number of equivalence relations on the set A = {1, 2, 3} is _______________ .
Mark the correct alternative in the following question:
Let L denote the set of all straight lines in a plane. Let a relation R be defined by lRm if l is perpendicular to m for all l, m ∈ L. Then, R is ______________ .
Let A = {1, 2, 3, 4}, B = {4, 5, 6}, C = {5, 6} Find (A × B) ∩ (A × C).
Let A = {0, 1, 2, 3} and define a relation R on A as follows: R = {(0, 0), (0, 1), (0, 3), (1, 0), (1, 1), (2, 2), (3, 0), (3, 3)}. Is R reflexive? symmetric? transitive?
Let L denote the set of all straight lines in a plane. Let a relation R be defined by lRm if and only if l is perpendicular to m ∀ l, m ∈ L. Then R is ______.
Consider the set A = {1, 2, 3} and the relation R = {(1, 2), (1, 3)}. R is a transitive relation.
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 R reflexive and transitive
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:
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 + y = 10, x, y ∈ N
Determine which of the above relations are reflexive, symmetric and transitive.
An integer m is said to be related to another integer n if m is a integral multiple of n. This relation in Z is reflexive, symmetric and transitive.
Let R be the relation on N defined as by x + 2 y = 8 The domain of R is ____________.
Let A = {1, 2, 3} and R = {(1, 2), (2, 3), (1, 3)} be a relation on A. Then, R is ____________.
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 ____________.
Let S = {1, 2, 3, 4, 5} and let A = S x S. Define the relation R on A as follows:
(a, b) R (c, d) iff ad = cb. Then, R is ____________.
Let the relation R in the set A = {x ∈ Z : 0 ≤ x ≤ 12}, given by R = {(a, b) : |a – b| is a multiple of 4}. Then [1], the equivalence class containing 1, is:
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?
