Let A be a finite set containing 3 elements, then the number of functions from A to A is

Options

A. 512
B. 511
C. 27
D. 26

Answer

To find the number of functions from a finite set \( A \) containing 3 elements to itself, we can follow these steps:

### Step 1: Identify the set
Let \( A = \{ a, b, c \} \). This set contains 3 elements.

### Step 2: Determine the mapping
We need to find the number of possible functions from set \( A \) to set \( A \). A function maps each element in the domain (the first set) to an element in the codomain (the second set).

### Step 3: Count the choices for each element
For each element in set \( A \), we can map it to any of the 3 elements in set \( A \):
- The element \( a \) can be mapped to \( a \), \( b \), or \( c \) (3 choices).
- The element \( b \) can also be mapped to \( a \), \( b \), or \( c \) (3 choices).
- The element \( c \) can similarly be mapped to \( a \), \( b \), or \( c \) (3 choices).

### Step 4: Calculate the total number of functions
Since the choices for each element are independent, we multiply the number of choices:
\[ \text{Total number of functions} = 3 \times 3 \times 3 = 3^3 = 27 \]

### Conclusion
Thus, the number of functions from set \( A \) to itself is \( 27 \).

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