Express `{(x,y):x^(2)+y^(2)=25, "where" x,y inW}` as a set of ordered pairs.

Answer

To express the set \(\{(x,y): x^2 + y^2 = 25, \text{ where } x,y \in W\}\) as a set of ordered pairs, we will follow these steps:

### Step 1: Understand the equation
The equation \(x^2 + y^2 = 25\) represents a circle with a radius of 5 centered at the origin in the coordinate plane. We need to find all pairs \((x, y)\) that satisfy this equation, where both \(x\) and \(y\) are whole numbers (non-negative integers).

### Step 2: Identify possible values for \(x\)
Since \(x\) and \(y\) are whole numbers, the possible values for \(x\) can range from \(0\) to \(5\) (as \(5^2 = 25\) is the maximum value for \(x^2\)).

### Step 3: Calculate corresponding values for \(y\)
For each value of \(x\), we will calculate \(y\) using the equation \(y^2 = 25 - x^2\):

1. **For \(x = 0\)**:
\[ y^2 = 25 - 0^2 = 25 \implies y = 5 \]
Ordered pairs: \((0, 5)\) and \((5, 0)\)

2. **For \(x = 1\)**:
\[ y^2 = 25 - 1^2 = 24 \implies y = \sqrt{24} \text{ (not a whole number)} \]

3. **For \(x = 2\)**:
\[ y^2 = 25 - 2^2 = 21 \implies y = \sqrt{21} \text{ (not a whole number)} \]

4. **For \(x = 3\)**:
\[ y^2 = 25 - 3^2 = 16 \implies y = 4 \]
Ordered pairs: \((3, 4)\) and \((4, 3)\)

5. **For \(x = 4\)**:
\[ y^2 = 25 - 4^2 = 9 \implies y = 3 \]
Ordered pairs: \((4, 3)\) and \((3, 4)\) (already counted)

6. **For \(x = 5\)**:
\[ y^2 = 25 - 5^2 = 0 \implies y = 0 \]
Ordered pairs: \((5, 0)\) and \((0, 5)\) (already counted)

### Step 4: Compile the unique ordered pairs
The unique ordered pairs that satisfy the equation \(x^2 + y^2 = 25\) are:
\[ \{(0, 5), (5, 0), (3, 4), (4, 3)\} \]

### Final Answer
Thus, the set of ordered pairs is:
\[ \{(0, 5), (5, 0), (3, 4), (4, 3)\} \]

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