Advertisements
Advertisements
प्रश्न
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
Advertisements
उत्तर
Given N = set of natural numbers
R is the relation defined by a R b if 2a + 3b = 30
3b = 30 – 2a ⇒ b = `(30 - 2a)/3` a, b ∈ N
a = 1, b = `(30 - 2)/3 = 28/3 ∉ "N"`
a = 2, b = `(30 - 4)/3 = 26/3 ∉ "N"`
a = 3, b = `(30 - 6)/3 = 24/3` = 8 ∈ N
∴ (3, 8) ∈ R
a = 4, b = `(30 - 8)/3 = 22/3 ∉ "N"`
a = 5, b = `(30 - 10)/3 = 20/3 ∉ "N"`
a = 6, b = `(30 - 12)/3 = 18/3` = 6 ∈ N
∴ (6, 6) ∈ R
a = 7, b = `(30 - 14)/3 = 16/3 ∉ "N"`
a = 8, b = `(30 - 16)/3 = 14/3 ∉ "N"`
a = 9, b = `(30 - 18)/3 = 12/3` = 4 ∈ N
∴ (9, 4) ∈ R
a = 10, b = `(30 - 20)/3 = 10/3 ∉ "N"`
a = 11, b = `(30 - 22)/3 = 8/3 ∉ "N"`
a = 12, b = `(30 - 24)/3 = 6/3` = 2 ∈ N
∴ (12, 2) ∈ R
a = 13, b = `(30 - 26)/3 = 4/3 ∉ "N"`
a = 14, b = `(30 - 28)/3 = 2/3 ∉ "N"`
a = 15, b = `(30 - 30)/3 = 0/3` = 0 ∈ N
When a > 15, b negative and does not belong to N.
∴ R = {(3, 8), (6, 6), (9, 4), (12, 2)}.
Clearly R is transitive since we cannot find elements (a, b), (b, c) in R such that (a, c) ∉ R
APPEARS IN
संबंधित प्रश्न
Define a relation R on the set N of natural numbers by R = {(x, y): y = x + 5, x is a natural number less than 4; x, y ∈ N}. Depict this relationship using roster form. Write down the domain and the range.
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 = (x, y, z) and B = (a, b). Find the total number of relations from A into B.
Let R be a relation from N to N defined by R = {(a, b) : a, b ∈ N and a = b2}. Is the statement true?
(a, b) ∈ R implies (b, a) ∈ R
Justify your answer in case.
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
Show that:
(i) (a, b) R (a, b) for all (a, b) ∈ N × N
If R = {(x, y) : x, y ∈ Z, x2 + y2 ≤ 4} is a relation on Z, then the domain of R is ______.
R is a relation from [11, 12, 13] to [8, 10, 12] defined by y = x − 3. Then, R−1 is
If P = {1, 2, 3) and Q = {1, 4}, find sets P × Q and Q × P
Let A = {1, 2, 3, 4), B = {4, 5, 6}, C = {5, 6}. Verify, A × (B ∪ C) = (A × B) ∪ (A × C)
Write the relation in the Roster Form. State its domain and range
R4 = {(x, y)/y > x + 1, x = 1, 2 and y = 2, 4, 6}
Write the relation in the Roster Form. State its domain and range
R7 = {(a, b)/a, b ∈ N, a + b = 6}
Identify which of if the following relations are reflexive, symmetric, and transitive.
| Relation | Reflexive | Symmetric | Transitive |
| R = {(a, b) : a, b ∈ Z, a – b is an integer} | |||
| R = {(a, b) : a, b ∈ N, a + b is even} | √ | √ | x |
| R = {(a, b) : a, b ∈ N, a divides b} | |||
| R = {(a, b) : a, b ∈ N, a2 – 4ab + 3b2 = 0} | |||
| R = {(a, b) : a is sister of b and a, b ∈ G = Set of girls} | |||
| R = {(a, b) : Line a is perpendicular to line b in a plane} | |||
| R = {(a, b) : a, b ∈ R, a < b} | |||
| R = {(a, b) : a, b ∈ R, a ≤ b3} |
Select the correct answer from given alternative.
A relation between A and B is
Select the correct answer from given alternative.
If (x, y) ∈ R × R, then xy = x2 is a relation which is
Answer the following:
Determine the domain and range of the following relation.
R = {(a, b)/a ∈ N, a < 5, b = 4}
Answer the following:
Check if R : Z → Z, R = {(a, b)/2 divides a – b} is equivalence relation.
Represent the given relation by
(a) an arrow diagram
(b) a graph and
(c) a set in roster form, wherever possible
{(x, y) | y = x + 3, x, y are natural numbers < 10}
Let P be the set of all triangles in a plane and R be the relation defined on P as aRb if a is similar to b. Prove that R is an equivalence relation
Choose the correct alternative:
Let f : R → R be defined by f(x) = 1 − |x|. Then the range of f is
Is the following relation a function? Justify your answer
R1 = `{(2, 3), (1/2, 0), (2, 7), (-4, 6)}`
