Let `A = {phi , {phi, {phi}} , {phi , {phi}, {phi, {phi}}}}`, then total number of relations on A that are not function is equal to

Options

A. 27
B. 512
C. 485
D. 1024

Explanation

To solve the problem, we need to find the total number of relations on the set \( A \) that are not functions. Let's go through the steps systematically.

### Step 1: Identify the Set \( A \)
The set \( A \) is given as:
\[ A = \{ \emptyset, \{ \emptyset \}, \{ \emptyset, \{ \emptyset \} \} \} \]

From this, we can see that the elements of \( A \) are:
1. \( \emptyset \)
2. \( \{ \emptyset \} \)
3. \( \{ \emptyset, \{ \emptyset \} \} \)

Thus, the cardinality of \( A \) (denoted as \( |A| \)) is 3.

### Step 2: Calculate the Total Number of Relations on \( A \)
A relation on a set \( A \) is a subset of the Cartesian product \( A \times A \). The number of elements in \( A \times A \) is given by:
\[ |A \times A| = |A| \times |A| = 3 \times 3 = 9 \]

The total number of relations on \( A \) is then given by:
\[ \text{Total Relations} = 2^{|A \times A|} = 2^9 = 512 \]

### Step 3: Calculate the Total Number of Functions from \( A \) to \( A \)
A function from set \( A \) to set \( A \) is defined by choosing an output in \( A \) for each input in \( A \). Therefore, the total number of functions is:
\[ \text{Total Functions} = |A|^{|A|} = 3^3 = 27 \]

### Step 4: Calculate the Total Number of Relations that are Not Functions
To find the total number of relations that are not functions, we subtract the number of functions from the total number of relations:
\[ \text{Relations that are not Functions} = \text{Total Relations} - \text{Total Functions} \]
\[ = 512 - 27 = 485 \]

### Final Answer
Thus, the total number of relations on \( A \) that are not functions is:
\[ \boxed{485} \]