To solve this problem, let's break it down step by step.

Step 1: Understand the Cartesian Product P × P

The Cartesian product $P \times P$ represents the set of all ordered pairs $(x, y)$ where $x \in P$ and $y \in P$. According to the problem, $P \times P$ has 9 elements, meaning $|P \times P| = 9$.

Step 2: Determine the Set P

If $P \times P$ has 9 elements, then the number of elements in $P$, denoted as $|P|$, must satisfy the equation:

$|P \times P| = |P| \times |P| = 9$

This implies:

$|P|^2 = 9 \quad \Rightarrow \quad |P| = 3$

So, the set $P$ contains 3 elements.

Step 3: Analyze Given Elements of $P \times P$

Among the 9 elements of $P \times P$ are the pairs $(-a, 0)$ and $(a, 0)$. This suggests that $P$ contains the elements $-a$, $0$, and $a$, because both $-a$ and $a$ are paired with 0.

Therefore, the set $P$ can be represented as:

$P = \{-a, 0, a\}$

Step 4: Analyze the Relation R

The relation $R$ is defined as:

$R = \{(x, y) : x + y = 0\}$

For each $x$ in $P$, there must be a corresponding $y$ in $P$ such that $x + y = 0$.

Given $P = \{-a, 0, a\}$:

• For $x = 0$, $y = 0$ satisfies the equation $0 + 0 = 0$.
• For $x = a$, $y = -a$ satisfies the equation $a + (-a) = 0$.
• For $x = -a$, $y = a$ satisfies the equation $-a + a = 0$.

Thus, the pairs $(0, 0)$, $(a, -a)$, and $(-a, a)$ are all in the relation $R$.

Conclusion

The set $P$ that satisfies all the given conditions is:

$\boxed{P = \{-a, 0, a\}}$

This is the required set $P$.

Would you like further details or have any additional questions? Here are some related questions you might be interested in:

1. What is a Cartesian product and how is it used in set theory?
2. How do you define a relation on a set?
3. Can a Cartesian product contain the same elements multiple times?
4. What is the difference between a relation and a function?
5. How do you represent a relation using a matrix?

Tip: When working with relations, always check if they satisfy the necessary properties, such as reflexivity, symmetry, and transitivity.