Let A = {(x, y) : 2x + 3y = 23, x, y ∈ N} and B = {x : (x, y) ∈ A}. Then the number of one-one funct

Q: Wait, why do we reject the negative root or non-integer values?

The problem explicitly states $x, y \in \mathbb{N}$. In the context of JEE mathematics, the set of natural numbers $\mathbb{N}$ is defined as $\{1, 2, 3, \dots\}$. Any solution that results in a fraction, zero, or a negative number violates the domain constraint given in the problem statement.

Q: Why is Set B defined only by the x-coordinates?

The problem defines $B = \{x : (x, y) \in A\}$. This is standard set-builder notation for the projection of the set $A$ onto the $x$-axis. It simply means we collect all the first elements of the ordered pairs found in $A$.

Q: Why is the answer $4!$ and not $4^4$?

Great question! $4^4$ would be the total number of functions from $A$ to $B$ (where each element in $A$ can map to any element in $B$). However, we are looking for one-one (injective) functions. For a one-one function, each element in $A$ must map to a *distinct* element in $B$. Since $|A| = |B| = 4$, this is equivalent to arranging 4 distinct items in 4 distinct slots, which is $4!$ or $^4 P_4$.

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 _______

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Imagine you are standing before a coordinate plane, looking at the line $2x + 3y = 23$. In a typical algebra class, this line is an infinite collection of points.

But here, we have a secret key: the constraint $x, y \in \mathbb{N}$. This transforms an infinite line into a discrete, finite set of points.

It is like finding hidden gems in a vast landscape. We are not just solving an equation; we are hunting for specific integer coordinates that satisfy the rigid laws of natural numbers.

### The Diophantine Hunt

To find these points, we rearrange our equation to isolate $y$:

$$y = \frac{23 - 2x}{3}$$

For $y$ to be a natural number, the numerator $23 - 2x$ must be a multiple of $3$. Let us test the smallest natural numbers for $x$.

If $x = 1$, then:

$$y = \frac{23 - 2}{3} = \frac{21}{3} = 7$$

This gives us our first point: $(1, 7)$.

If we try $x = 2$, $y = \frac{19}{3}$ (not an integer). If we try $x = 3$, $y = \frac{17}{3}$ (not an integer).

But look at $x = 4$:

$$y = \frac{23 - 8}{3} = \frac{15}{3} = 5$$

We have found $(4, 5)$. Notice the pattern? The $x$-values are increasing by $3$ each time!

This is the elegance of linear equations. Continuing this, we find $x = 7 \implies y = 3$ and $x = 10 \implies y = 1$.

If we go to $x = 13$, $y$ becomes negative, which breaks our natural number rule. Thus, our set $A$ is defined as $A = \{(1, 7), (4, 5), (7, 3), (10, 1)\}$.

### Constructing the Sets

With Set $A$ firmly in our grasp, defining Set $B$ is a breeze. The problem tells us $B = \{x : (x, y) \in A\}$.

We simply extract the $x$-coordinates from our pairs in $A$. Thus, $B = \{1, 4, 7, 10\}$.

Both sets have exactly $4$ elements. We have successfully mapped the geometry of the line to the discrete world of sets.

### The Combinatorial Bridge

Now, we reach the final act: counting the one-one functions from $A$ to $B$. A one-one function is a mapping where every element in the domain $A$ is assigned to a unique element in the codomain $B$.

Since both sets have $4$ elements, we are essentially asking: in how many ways can we pair up $4$ unique inputs with $4$ unique outputs? This is the definition of a permutation.

The number of ways to arrange $4$ items is $4!$, which is calculated as:

$$4! = 4 	imes 3 	imes 2 	imes 1 = 24$$

The logic is simple, yet profound. We have navigated from a linear equation to a set, and finally to a combinatorial result.

The final answer is **24**.

Animated Solution for Mathematics - Permutations and Combinations: Let $A = {(x, y) : 2 x + 3 y = 23, x, y \in N}$ and $B = {x : (x, y) \in A}$ . Then the number of one-one functions from $A$ to $B$ is equal to _______

Enter Numerical Value:

Visualized Solution

Analyzing the Equation $2 x + 3 y = 23$

Isolating $y$ for Calculation

Testing $x = 1$

Finding the First Solution $(1, 7)$

Finding the Second Solution $(4, 5)$

Finding the Remaining Solutions

Defining Set $A$

Defining Set $B$

Counting One-One Functions

Final Calculation

The Sigma Insight: Linear Permutations

Analyzing the Setup

The Diophantine Hunt

Constructing the Sets

The Combinatorial Bridge

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