If `P(S)` denotes the set of all subsets of a given set S, then the number of one-to-one functions from the set `S= {1,2,3}` to the set `P(S)` is

Options

A. 8
B. 320
C. 336
D. 24

Explanation

To solve the problem of finding the number of one-to-one functions from the set \( S = \{1, 2, 3\} \) to the power set \( P(S) \), we can follow these steps:

### Step 1: Determine the elements of the set \( S \)
The set \( S \) is given as \( S = \{1, 2, 3\} \). This set contains 3 elements.

### Step 2: Calculate the power set \( P(S) \)
The power set \( P(S) \) is the set of all subsets of \( S \). The number of subsets of a set with \( n \) elements is given by \( 2^n \).

Here, since \( n = 3 \):
\[ |P(S)| = 2^3 = 8 \]

The subsets of \( S \) (elements of \( P(S) \)) are:
1. \( \emptyset \)
2. \( \{1\} \)
3. \( \{2\} \)
4. \( \{3\} \)
5. \( \{1, 2\} \)
6. \( \{1, 3\} \)
7. \( \{2, 3\} \)
8. \( \{1, 2, 3\} \)

### Step 3: Understand the requirement for one-to-one functions
A one-to-one function (injective function) from set \( S \) to set \( P(S) \) means that each element of \( S \) must map to a unique element in \( P(S) \). No two elements in \( S \) can map to the same element in \( P(S) \).

### Step 4: Count the number of one-to-one functions
To find the number of one-to-one functions from \( S \) to \( P(S) \):
- The first element of \( S \) (let's say 1) can be mapped to any of the 8 elements in \( P(S) \).
- The second element of \( S \) (2) can then be mapped to any of the remaining 7 elements in \( P(S) \) (since one element is already taken).
- The third element of \( S \) (3) can be mapped to any of the remaining 6 elements in \( P(S) \).

Thus, the total number of one-to-one functions is calculated as follows:
\[ \text{Total one-to-one functions} = 8 \times 7 \times 6 \]

Calculating this gives:
\[ 8 \times 7 = 56 \]
\[ 56 \times 6 = 336 \]

### Conclusion
The number of one-to-one functions from the set \( S \) to the power set \( P(S) \) is \( 336 \).