Let set P = {a, b, {b}, {{a}, 3}}. Which one of the following is NOT a valid relation on P?

Select one:

a.

{(b, {b})}

b.

{({a}, 3), (a, b), ({b}, {b})}

c.

{(a, a), (a, b), ({{a}, 3}, {b})}

d.

{({{a}, 3}, {{a}, 3})}

To determine which of the given options is **not** a valid relation on set \( P = \{a, b, \{b\}, \{\{a\}, 3\}\} \), we need to understand what constitutes a valid relation on \( P \).

### Definition:
A relation on a set \( P \) is a subset of the Cartesian product \( P \times P \). This means that any relation on \( P \) must consist of ordered pairs where both elements in each pair belong to \( P \).

### Step 1: Identify the elements of \( P \):
\[ P = \{a, b, \{b\}, \{\{a\}, 3\}\} \]

### Step 2: Determine the Cartesian product \( P \times P \):
The Cartesian product \( P \times P \) consists of all ordered pairs \( (x, y) \) where \( x \) and \( y \) are elements of \( P \).

Possible pairs in \( P \times P \) are:
\[
\begin{aligned}
& (a, a), (a, b), (a, \{b\}), (a, \{\{a\}, 3\}), \\
& (b, a), (b, b), (b, \{b\}), (b, \{\{a\}, 3\}), \\
& (\{b\}, a), (\{b\}, b), (\{b\}, \{b\}), (\{b\}, \{\{a\}, 3\}), \\
& (\{\{a\}, 3\}, a), (\{\{a\}, 3\}, b), (\{\{a\}, 3\}, \{b\}), (\{\{a\}, 3\}, \{\{a\}, 3\})
\end{aligned}
\]

### Step 3: Analyze each option:
Now, let's check if each option is a subset of \( P \times P \).

- **Option a: \( \{(b, \{b\})\} \)**  
  - \( (b, \{b\}) \) is a valid pair, where both elements \( b \) and \( \{b\} \) are in \( P \).
  - This is a valid relation.

- **Option b: \( \{(\{a\}, 3), (a, b), (\{b\}, \{b\})\} \)**  
  - \( (a, b) \) is valid since both \( a \) and \( b \) are in \( P \).
  - \( (\{b\}, \{b\}) \) is valid since \( \{b\} \) is in \( P \).
  - **\( (\{a\}, 3) \) is not valid** because \( \{a\} \) and \( 3 \) are not individual elements of \( P \) (they are part of \( \{\{a\}, 3\} \), but not as separate elements).  
  - This is **not** a valid relation.

- **Option c: \( \{(a, a), (a, b), (\{\{a\}, 3\}, \{b\})\} \)**  
  - All pairs \( (a, a) \), \( (a, b) \), and \( (\{\{a\}, 3\}, \{b\}) \) are valid since each element is in \( P \).
  - This is a valid relation.

- **Option d: \( \{(\{\{a\}, 3\}, \{\{a\}, 3\})\} \)**  
  - \( (\{\{a\}, 3\}, \{\{a\}, 3\}) \) is valid since \( \{\{a\}, 3\} \) is in \( P \).
  - This is a valid relation.

### Conclusion:
**Option b** is not a valid relation on \( P \) because it includes the pair \( (\{a\}, 3) \), where neither \( \{a\} \) nor \( 3 \) are elements of \( P \). 

Thus, the correct answer is **Option b**. 

---

Would you like more details or have any further questions? Here are some additional questions to consider:

1. What is the Cartesian product of two sets, and how is it computed?
2. How does one determine if a set of ordered pairs is a valid relation on a given set?
3. What are some examples of relations that are not valid, and why?
4. Can a relation contain pairs where elements are not in the original set? Why or why not?
5. What is the difference between a relation and a function?

**Tip:** When determining if a pair is a valid element of a relation, always check if both elements are members of the original set.

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