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

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