The cartesign product P×P has 9 elements among which are (-a,0)&(a,0) A relation from P to P is defined as R={(x,y):x+y=0} Find P

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.

Explore the Cartesian product P × P with 9 elements and the relation R={(x,y):x+y=0}. Learn how to determine the set P and analyze the relation based on given elements (-a,0) and (a,0). Suitable for advanced high school students studying set theory.

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