The number of reflexive relations of a set with four elements is equal to

Options

A. `2 ^(16)`
B. `2 ^(12)`
C. `2 ^(8)`
D. `2 ^(4)`

Answer

To find the number of reflexive relations of a set with four elements, we can follow these steps:

### Step 1: Understand Reflexive Relations
A reflexive relation on a set is a relation where every element is related to itself. For a set with \( n \) elements, this means that each element \( a_i \) must satisfy the relation \( (a_i, a_i) \).

### Step 2: Determine the Number of Pairs
For a set with \( n \) elements, the total number of ordered pairs that can be formed is \( n^2 \). For our case, with \( n = 4 \):
\[ n^2 = 4^2 = 16 \]
This means there are 16 possible ordered pairs in total.

### Step 3: Identify Mandatory Pairs
In a reflexive relation, the pairs \( (a_1, a_1), (a_2, a_2), (a_3, a_3), (a_4, a_4) \) must be included. There are 4 such mandatory pairs for our set of 4 elements.

### Step 4: Determine Remaining Pairs
After including the mandatory pairs, we have:
\[ 16 \text{ (total pairs)} - 4 \text{ (mandatory pairs)} = 12 \text{ (remaining pairs)} \]
These remaining pairs can either be included in the relation or not.

### Step 5: Calculate the Number of Reflexive Relations
For each of the 12 remaining pairs, we have 2 choices: either include the pair in the relation or exclude it. Thus, the total number of reflexive relations can be calculated as:
\[ 2^{12} \]

### Conclusion
The number of reflexive relations of a set with four elements is \( 2^{12} \).

### Final Answer
The number of reflexive relations of a set with four elements is equal to \( 2^{12} \).

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