Question

Let \[ A = \{(x, y) : 2x + 3y = 23, \, x, y \in \mathbb{N}\} \] and \[ B = \{x : (x, y) \in A\}. \] Then the number of one-one functions from \( A \) to \( B \) is equal to _________ .

Answer 1

Correct Answer: 24

To resolve this issue, we first determine the elements of the set \( A = \{(x, y) : 2x + 3y = 23, \, x, y \in \mathbb{N}\} \). This involves finding pairs \((x, y)\) that satisfy the equation \(2x + 3y = 23\), where \(x\) and \(y\) are natural numbers. We will examine potential \(x\) values and compute their corresponding \(y\) values.

Step 1: As \(x\) and \(y\) are natural numbers, they are positive integers. The smallest possible value for \(x\) is 1.

Step 2: Calculate potential values. We derive \(y\) from the equation:

\(2x + 3y = 23 \Rightarrow 3y = 23 - 2x\). For \(y\) to be an integer, \((23 - 2x)\) must be divisible by 3.

Step 3: Test \(x\) values:

x	23 - 2x	Divisible by 3?	y
1	21	Yes	7
2	19	No	-
3	17	No	-
4	15	Yes	5
5	13	No	-
6	11	No	-
7	9	Yes	3
8	7	No	-
9	5	No	-
10	3	Yes	1

Step 4: The set \( A \) comprises the pairs: \((1,7), (4,5), (7,3), (10,1)\). Therefore, \(A = \{(1, 7), (4, 5), (7, 3), (10, 1)\}\) and \(|A| = 4\).

Step 5: Define set \( B = \{x : (x, y) \in A\}\). Consequently, \( B = \{1, 4, 7, 10\} \). The cardinality of \( B \), denoted \(|B|\), is 4.

Step 6: Compute the number of one-one functions from \( A \) to \( B \). A one-one (bijective) function can exist only if \(|A| = |B|\). The count of such functions equals the number of permutations of \(|B|\), which is \(4!\).

\[4! = 24\]

Conclusion: The total number of one-one functions from \( A \) to \( B \) is 24, which falls within the specified range of 24 to 24.