Advertisements
Advertisements
प्रश्न
Answer the following:
Show that the following is an equivalence relation
R in A = {x ∈ Z | 0 ≤ x ≤ 12} given by R = {(a, b)/|a − b| is a multiple of 4}
उत्तर
A = {x ∈ Z | 0 ≤ x ≤ 12}
R = {(a, b)/|a − b| is a multiple of 4; a, b ∈ A}
|a − a| = 0 is a multiple of 4
∴ aRa ∀ a∈A
∴ R is reflexive
Let aRb
∴ |a − b| is a multiple of 4
∴ |b − a| = |a − b|
∴ |b − a| is a multiple of 4
∴ aRb ⇒ bRa ∀a, b ∈ A
∴ R is symmetric
Let aRb and bRc
∴ |a − b| and |b − c| are multiples of 4
∴ a − b = 4m, b − c = 4n; m, n ∈ Z
a − c = (a − b) + (b − c) = 4m + 4n
= 4(m + n); (m + n) ∈ Z
∴ |a − c| is a multiple of 4
∴ aRc
∴ aRb, bRc ⇒ aRc ∀a, b, c ∈ A
∴ R is transitive
∵ R is reflexive, symmetric and transitive
∴ R is an equivalence relation.
APPEARS IN
संबंधित प्रश्न
Let A = {x, y, z} and B = {1, 2}. Find the number of relations from A to B.
The relation f is defined by f(x) = `{(x^2,0<=x<=3),(3x,3<=x<=10):}`
The relation g is defined by g(x) = `{(x^2, 0 <= x <= 2),(3x,2<= x <= 10):}`
Show that f is a function and g is not a function.
Determine the domain and range of the relation R defined by
(ii) R = {(x, x3) : x is a prime number less than 10}
Determine the domain and range of the relations:
(i) R = {(a, b) : a ∈ N, a < 5, b = 4}
Define a relation R on the set N of natural number by R = {(x, y) : y = x + 5, x is a natural number less than 4, x, y ∈ N}. Depict this relationship using (i) roster form (ii) an arrow diagram. Write down the domain and range or R.
Let R be a relation on N × N defined by
(a, b) R (c, d) ⇔ a + d = b + c for all (a, b), (c, d) ∈ N × N
(iii) (a, b) R (c, d) and (c, d) R (e, f) ⇒ (a, b) R (e, f) for all (a, b), (c, d), (e, f) ∈ N × N
If R is a relation from set A = (11, 12, 13) to set B = (8, 10, 12) defined by y = x − 3, then write R−1.
Let R = [(x, y) : x, y ∈ Z, y = 2x − 4]. If (a, -2) and (4, b2) ∈ R, then write the values of a and b.
If A = [1, 3, 5] and B = [2, 4], list of elements of R, if
R = {(x, y) : x, y ∈ A × B and x > y}
Let A = {6, 8} and B = {1, 3, 5}
Show that R1 = {(a, b)/a ∈ A, b ∈ B, a − b is an even number} is a null relation. R2 = {(a, b)/a ∈ A, b ∈ B, a + b is odd number} is an universal relation
Write the relation in the Roster Form. State its domain and range
R3 = {(x, y)/y = 3x, y∈ {3, 6, 9, 12}, x∈ {1, 2, 3}
Write the relation in the Roster Form. State its domain and range
R8 = {(a, b)/b = a + 2, a ∈ z, 0 < a < 5}
Select the correct answer from given alternative.
The relation ">" in the set of N (Natural number) is
Answer the following:
R = {1, 2, 3} → {1, 2, 3} given by R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 3)} Check if R is reflexive
Answer the following:
R = {1, 2, 3} → {1, 2, 3} given by R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 3)} Check if R is symmentric
Answer the following:
R = {1, 2, 3} → {1, 2, 3} given by R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 3)} Check if R is transitive
Let A = {1, 2, 3, 7} and B = {3, 0, –1, 7}, the following is relation from A to B?
R1 = {(2, 1), (7, 1)}
Let A = {1, 2, 3, 7} and B = {3, 0, –1, 7}, the following is relation from A to B?
R3 = {(2, –1), (7, 7), (1, 3)}
Represent the given relation by
(a) an arrow diagram
(b) a graph and
(c) a set in roster form, wherever possible
{(x, y) | x = 2y, x ∈ {2, 3, 4, 5}, y ∈ {1, 2, 3, 4}
Multiple Choice Question :
If there are 1024 relation from a set A = {1, 2, 3, 4, 5} to a set B, then the number of elements in B is
Discuss the following relation for reflexivity, symmetricity and transitivity:
Let A be the set consisting of all the female members of a family. The relation R defined by “aRb if a is not a sister of b”
Let X = {a, b, c, d} and R = {(a, a), (b, b), (a, c)}. Write down the minimum number of ordered pairs to be included to R to make it symmetric
Let X = {a, b, c, d} and R = {(a, a), (b, b), (a, c)}. Write down the minimum number of ordered pairs to be included to R to make it transitive
Let A = {a, b, c} and R = {(a, a), (b, b), (a, c)}. Write down the minimum number of ordered pairs to be included to R to make it equivalence
On the set of natural numbers let R be the relation defined by aRb if 2a + 3b = 30. Write down the relation by listing all the pairs. Check whether it is transitive
Prove that the relation “friendship” is not an equivalence relation on the set of all people in Chennai
On the set of natural numbers let R be the relation defined by aRb if a + b ≤ 6. Write down the relation by listing all the pairs. Check whether it is symmetric
On the set of natural numbers let R be the relation defined by aRb if a + b ≤ 6. Write down the relation by listing all the pairs. Check whether it is transitive
Choose the correct alternative:
Let R be the set of all real numbers. Consider the following subsets of the plane R × R: S = {(x, y) : y = x + 1 and 0 < x < 2} and T = {(x, y) : x − y is an integer} Then which of the following is true?
Choose the correct alternative:
Let f : R → R be defined by f(x) = 1 − |x|. Then the range of f is
Given R = {(x, y) : x, y ∈ W, x2 + y2 = 25}. Find the domain and Range of R.
If R3 = {(x, x) | x is a real number} is a relation. Then find domain and range of R3.
Is the given relation a function? Give reasons for your answer.
f = {(x, x) | x is a real number}
Let S = {x ∈ R : x ≥ 0 and `2|sqrt(x) - 3| + sqrt(x)(sqrt(x) - 6) + 6 = 0}`. Then S ______.
Let A = {1, 2, 3, 4}, B = {1, 5, 9, 11, 15, 16} and f = {(1, 5), (2, 9), (3, 1), (4, 5), (2, 11)}. Is the following true?
f is a function from A to B
Justify your answer in case.
