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 solve this problem, we first identify the elements of the set \( A = \{(x, y) : 2x + 3y = 23, \, x, y \in \mathbb{N}\} \). We need to find pairs \((x, y)\) such that they satisfy the equation \(2x + 3y = 23\) with \(x, y\) being natural numbers. Let's explore possible values of \(x\) and determine corresponding \(y\) values.

Step 1: Since \(x\) and \(y\) are natural numbers, they must be positive integers. Therefore, the minimum value for \(x\) is 1.

Step 2: Calculate possible values. We find values of \(y\) by manipulating the equation:

\(2x + 3y = 23 \Rightarrow 3y = 23 - 2x\). \(y\) is an integer if \((23 - 2x)\) is divisible by 3.

Step 3: Check values of \(x\):

x	23 - 2x	Is it 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 \( A \) consists of these pairs: \((1,7), (4,5), (7,3), (10,1)\). Thus, \(A = \{(1, 7), (4, 5), (7, 3), (10, 1)\}\) and \(|A| = 4\).

Step 5: Set \( B = \{x : (x, y) \in A\}\). Consequently, \( B = \{1, 4, 7, 10\} \). Therefore, the cardinality of \( B \), \(|B| = 4\).

Step 6: Calculate the number of one-one functions from \( A \) to \( B \). A one-one (bijective) function exists when \(|A| = |B|\). The number of such functions is equal to the number of permutations of \(|B|\), which are represented as \(4!\).

\[4! = 24\]

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

Answer 2

We are given that:

\( A = \{(x, y) : 2x + 3y = 23, \, x, y \in \mathbb{N} \} \)

\( B = \{x : (x, y) \in A\} \)

We need to find the number of one-to-one functions from \( A \) to \( B \).

Step 1: Find the Elements of Set A
We are given the equation \( 2x + 3y = 23 \), where \( x \) and \( y \) are natural numbers (\( \mathbb{N} \)).
To solve for \( y \) in terms of \( x \), we rearrange the equation:

\( 3y = 23 - 2x \Rightarrow y = \frac{23 - 2x}{3} \)

For \( y \) to be a natural number, \( 23 - 2x \) must be divisible by 3. Thus, we need to solve the congruence:

\( 23 - 2x \equiv 0 \pmod{3} \)

Simplifying:
\( 23 \equiv 2 \pmod{3} \) and \( 2x \equiv 2 \pmod{3} \)
\( x \equiv 1 \pmod{3} \)

Thus, \( x \) must be of the form \( x = 3k + 1 \) for some integer \( k \). Now, let’s substitute values of \( x \) into the equation \( 2x + 3y = 23 \) and solve for \( y \).

For \( x = 1 \):
\( 2(1) + 3y = 23 \Rightarrow 2 + 3y = 23 \Rightarrow 3y = 21 \Rightarrow y = 7 \)

Thus, \( (x, y) = (1, 7) \).

For \( x = 4 \):
\( 2(4) + 3y = 23 \Rightarrow 8 + 3y = 23 \Rightarrow 3y = 15 \Rightarrow y = 5 \)
Thus, \( (x, y) = (4, 5) \).

For \( x = 7 \):
\( 2(7) + 3y = 23 \Rightarrow 14 + 3y = 23 \Rightarrow 3y = 9 \Rightarrow y = 3 \)
Thus, \( (x, y) = (7, 3) \).

For \( x = 10 \):
\( 2(10) + 3y = 23 \Rightarrow 20 + 3y = 23 \Rightarrow 3y = 3 \Rightarrow y = 1 \)
Thus, \( (x, y) = (10, 1) \).

So, the elements of set \( A \) are:

\( A = \{(1, 7), (4, 5), (7, 3), (10, 1)\} \)

Step 2: Define Set \( B \)
Set \( B = \{x : (x, y) \in A\} \). Thus, \( B = \{1, 4, 7, 10\} \).

Step 3: Find the Number of One-to-One Functions
The number of one-to-one functions from \( A \) to \( B \) is the number of ways to assign each element of \( A \) to a unique element of \( B \). Since both sets \( A \) and \( B \) contain 4 elements, the number of one-to-one functions is simply the number of permutations of 4 elements, which is:

\( 4! = 24 \)

Thus, the number of one-to-one functions from \( A \) to \( B \) is:

24