Set A contain 3 elements , set B contain 5 elements , number of one-one function from `A to B` is "x" and number of one-one functions from `A to AxxB` is "y" then relation between x and y

Options

A. `2y=78x`
B. `4y=91x`
C. `2y=91x`
D. `y=52x`

Answer

To solve the problem, we need to find the number of one-one functions from set A to set B and from set A to the Cartesian product of A and B. Let's denote the number of elements in set A as |A| and the number of elements in set B as |B|.

Given:
- Set A contains 3 elements, so |A| = 3.
- Set B contains 5 elements, so |B| = 5.

### Step 1: Calculate the number of one-one functions from A to B (denote this as x)
A one-one function (or injective function) from set A to set B means that each element in A must map to a unique element in B. The number of one-one functions can be calculated using the formula:

\[ x = |B| \times (|B| - 1) \times (|B| - 2) \]

Substituting the values:

\[ x = 5 \times 4 \times 3 = 60 \]

### Step 2: Calculate the number of elements in the Cartesian product A × B
The Cartesian product A × B consists of all possible ordered pairs (a, b) where \( a \in A \) and \( b \in B \). The number of elements in A × B is given by:

\[ |A \times B| = |A| \times |B| = 3 \times 5 = 15 \]

### Step 3: Calculate the number of one-one functions from A to A × B (denote this as y)
Now we need to find the number of one-one functions from set A to the Cartesian product A × B. Since A × B has 15 elements, the number of one-one functions from A to A × B is:

\[ y = |A \times B| \times (|A \times B| - 1) \times (|A \times B| - 2) \]

Substituting the values:

\[ y = 15 \times 14 \times 13 \]

### Step 4: Relate x and y
Now we have:

- \( x = 60 \)
- \( y = 15 \times 14 \times 13 \)

To simplify \( y \):

\[ y = 15 \times 14 \times 13 = 2730 \]

### Step 5: Establish the relation between x and y
We can express the relationship between \( x \) and \( y \):

\[ 2y = 91x \]

This means that \( y \) is directly proportional to \( x \) with a factor of \( \frac{91}{2} \).

### Final Relation
Thus, the relation between \( x \) and \( y \) can be summarized as:

\[ 2y = 91x \]