Let A be a set containing 10 distinct elements, then the total number of distinct functions from A to A is

Options

A. `10^(1)`
B. `10^(10)`
C. `2^(10)`
D. `2^(10) - 1`

Answer

To find the total number of distinct functions from set A to set A, where A contains 10 distinct elements, we can follow these steps:

### Step 1: Understand the problem
We need to determine how many distinct functions can be formed from a set A with 10 elements to itself.

### Step 2: Define the sets
Let A = {a1, a2, a3, a4, a5, a6, a7, a8, a9, a10}, where each ai represents a distinct element of the set A.

### Step 3: Determine the number of choices for each element
For each element in the domain (which is also set A), we can map it to any of the 10 elements in the codomain (also set A).

### Step 4: Calculate the total number of functions
Since there are 10 elements in the domain and each element has 10 choices in the codomain, the total number of distinct functions can be calculated using the formula:
\[ \text{Total functions} = (\text{Number of choices for the first element})^{(\text{Number of elements in the domain})} \]
This translates to:
\[ \text{Total functions} = 10^{10} \]

### Step 5: Conclusion
Thus, the total number of distinct functions from A to A is \(10^{10}\).

### Final Answer
The total number of distinct functions from set A to set A is \(10^{10}\).

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