Question

Let the relation R is defined in N by \(a \mathbb{R} b\), if \(3a+2b = 27\) , then R is

• A. {(1,12), (3,9), (5,6), (7,3)}
• B. {(1,12), (3,9), (5,6), (7,3), (9,0)}
• C. \(\{(0, \frac{27}{2}), (1,12), (3,9), (5,6), (7,3)\}\)
• D. {(2,1), (9,3), (6,5), (3,7)}

Answer 1

The Correct Option is A

To determine the relation R defined in N by a \(R_ b\), if \(3a + 2b = 27\), we need to find the pairs (a, b) that satisfy this equation.
Let's check each option to see which one represents the relation R.
Option (A): {(1, 12), (3, 9), (5, 6), (7, 3)}
Let's substitute the values from each pair into the equation:
\(3(1) + 2(12) = 3 + 24 = 27\) (satisfied)
\(3(3) + 2(9) = 9 + 18 = 27\) (satisfied)
\(3(5) + 2(6) = 15 + 12 = 27\) (satisfied)
\(3(7) + 2(3) = 21 + 6 = 27\) (satisfied)
So, option (A) represents the relation R.
Therefore, the relation R defined in N by a\( R_ b\), if 3a + 2b = 27, is {(1, 12), (3, 9), (5, 6), (7, 3)} (option A).

Answer 2

We are given a relation R defined in the set of natural numbers ℕ such that:

\( a \mathbb{R} b \iff 3a + 2b = 27 \) 

We need to find all pairs (a, b) in natural numbers such that the equation is satisfied.

Solving for b: \[ b = \frac{27 - 3a}{2} \]

We test values of a ∈ ℕ and check if b is also a natural number:

a	b = (27 - 3a)/2	b ∈ ℕ?
1	12	Yes
3	9	Yes
5	6	Yes
7	3	Yes
9	0	No (0 ∉ ℕ)

Hence, R = {(1,12), (3,9), (5,6), (7,3)}

Answer: {(1,12), (3,9), (5,6), (7,3)}

Answer 3

The relation R is defined in N (the set of natural numbers) by a R b if 3a + 2b = 27. We need to find the pairs (a, b) that satisfy this condition where both a and b are natural numbers (positive integers).

Let's rearrange the equation to solve for b:

2b = 27 - 3a

b = (27 - 3a) / 2

Now we need to find the values of 'a' for which 'b' is also a natural number. This means that (27 - 3a) must be positive and divisible by 2.

• If a = 1: b = (27 - 3(1)) / 2 = (27 - 3) / 2 = 24 / 2 = 12. (1, 12) is in R.
• If a = 2: b = (27 - 3(2)) / 2 = (27 - 6) / 2 = 21 / 2 = 10.5. This is not a natural number.
• If a = 3: b = (27 - 3(3)) / 2 = (27 - 9) / 2 = 18 / 2 = 9. (3, 9) is in R.
• If a = 4: b = (27 - 3(4)) / 2 = (27 - 12) / 2 = 15 / 2 = 7.5. This is not a natural number.
• If a = 5: b = (27 - 3(5)) / 2 = (27 - 15) / 2 = 12 / 2 = 6. (5, 6) is in R.
• If a = 6: b = (27 - 3(6)) / 2 = (27 - 18) / 2 = 9 / 2 = 4.5. This is not a natural number.
• If a = 7: b = (27 - 3(7)) / 2 = (27 - 21) / 2 = 6 / 2 = 3. (7, 3) is in R.
• If a = 8: b = (27 - 3(8)) / 2 = (27 - 24) / 2 = 3 / 2 = 1.5. This is not a natural number.
• If a = 9: b = (27 - 3(9)) / 2 = (27 - 27) / 2 = 0 / 2 = 0. 0 is NOT a natural number.
• 27-3a > 0
• 3a < 27
• a < 9

Therefore, the relation R = {(1, 12), (3, 9), (5, 6), (7, 3)}.

Answer: {(1,12), (3,9), (5,6), (7,3)}