Advertisements
Advertisements
प्रश्न
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
Advertisements
उत्तर
N = the set of natural numbers.
R is the relation defined on N by
a R b if a + b ≤ 6
R = {(a, b), a, b ∈ N / a + b ≤ 6}
a + b ≤ 6 ⇒ b ≤ 6 – a
a = 1,
b ≤ 6 – 1 = 5
b is 1, 2, 3, 4, 5
∴ (1, 1), (1, 2), (1, 3), (1, 4), (1, 5) ∈ R
a = 2,
b ≤ 6 – 2 = 4
b is 1, 2, 3, 4
∴ (2, 1), (2, 2), (2, 3), (2, 4) ∈ R
a = 3,
b < 6 – 3 = 3
b is 1, 2, 3
∴ (3, 1), (3, 2), (3, 3) ∈ R
a = 4 ,
b < 6 – 4 = 2
b is 1, 2
∴ (4, 1), (4, 2) ∈ R
a = 5,
b < 6 – 5 = 1
b is 1
∴ (5, 1) ∈ R
∴ R = {(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (2, 1), (2, 2), (2, 3), (2, 4), (3, 1), (3, 2), (3, 3), (4, 1), (4, 2), (5, 1)}
Symmetric:
Cleary R is symmetric forever (x, y) ∈ R, we have (y, x) ∈ R.
APPEARS IN
संबंधित प्रश्न
A = {1, 2, 3, 5} and B = {4, 6, 9}. Define a relation R from A to B by R = {(x, y): the difference between x and y is odd; x ∈ A, y ∈ B}. Write R in roster form.
Let R be the relation on Z defined by R = {(a, b): a, b ∈ Z, a – b is an integer}. Find the domain and range of R.
If A = [1, 2, 3], B = [4, 5, 6], which of the following are relations from A to B? Give reasons in support of your answer.
(i) [(1, 6), (3, 4), (5, 2)]
(ii) [(1, 5), (2, 6), (3, 4), (3, 6)]
(iii) [(4, 2), (4, 3), (5, 1)]
(iv) A × B.
Find the inverse relation R−1 in each of the cases:
(i) R = {(1, 2), (1, 3), (2, 3), (3, 2), (5, 6)}
Let A = (3, 5) and B = (7, 11). Let R = {(a, b) : a ∈ A, b ∈ B, a − b is odd}. Show that R is an empty relation from A into B.
The adjacent figure shows a relationship between the sets P and Q. Write this relation in (i) set builder form (ii) roster form. What is its domain and range?
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
Let R be a relation on N defined by x + 2y = 8. The domain of R is
If P = {1, 2, 3) and Q = {1, 4}, find sets P × Q and Q × P
Express {(x, y) / x2 + y2 = 100, where x, y ∈ W} as a set of ordered pairs
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
Select the correct answer from given alternative.
The relation ">" in the set of N (Natural number) is
Answer the following:
Find R : A → A when A = {1, 2, 3, 4} such that R = (a, b)/a − b = 10}
Answer the following:
Check if R : Z → Z, R = {(a, b)/2 divides a – b} is equivalence relation.
Multiple Choice Question :
Let n(A) = m and n(B) = n then the total number of non-empty relation that can be defined from A to B is ________.
Discuss the following relation for reflexivity, symmetricity and transitivity:
Let A be the set consisting of all the members of a family. The relation R defined by “aRb if a is not a sister of b”
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 reflexive
Find the domain and range of the relation R given by R = {(x, y) : y = `x + 6/x`; where x, y ∈ N and x < 6}.
Is the given relation a function? Give reasons for your answer.
g = `"n", 1/"n" |"n"` is a positive integer
