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 ________

We need to find natural number pairs $$(x, y)$$ that satisfy $$2x + 3y = 23$$. Since $$3y$$ must be odd (because $$23$$ is odd and $$2x$$ is even), $$y$$ must be odd.

• If $$y=1: 2x + 3 = 23 \implies 2x = 20 \implies x = 10$$. Pair: $$(10, 1)$$

• If $$y=3: 2x + 9 = 23 \implies 2x = 14 \implies x = 7$$. Pair: $$(7, 3)$$

• If $$y=5: 2x + 15 = 23 \implies 2x = 8 \implies x = 4$$. Pair: $$(4, 5)$$

• If $$y=7: 2x + 21 = 23 \implies 2x = 2 \implies x = 1$$. Pair: $$(1, 7)$$

• If $$y=9: 2x + 27 = 23$$ (No natural number solution).

So, $$A = \{(10, 1), (7, 3), (4, 5), (1, 7)\}$$. The number of elements $$n(A) = 4$$.

Finding the elements of Set $$B$$

$$B$$ consists of the $$x$$-coordinates from $$A$$.

$$B = \{10, 7, 4, 1\}$$. The number of elements $$n(B) = 4$$.

 Calculating One-One Functions

A one-one function (injection) from a set of size $$n$$ to a set of size $$n$$ is simply a permutation of the elements.

$$\text{Number of functions} = n! = 4! = 4 \times 3 \times 2 \times 1 = 24$$