To find the diameter of the subset \( S \) defined by the conditions given in the question, we will follow these steps:
### Step 1: Understand the constraints defining the set \( S \)
The set \( S \) is defined by the following inequalities:
1. \( y - x \leq 0 \) (or \( y \leq x \))
2. \( x + y \geq 0 \) (or \( y \geq -x \))
3. \( x^2 + y^2 \leq 2 \) (which describes a circle with radius \( \sqrt{2} \))
### Step 2: Graph the inequalities
1. **Graph \( y = x \)**: This is a line passing through the origin at a \( 45^\circ \) angle.
2. **Graph \( y = -x \)**: This is another line passing through the origin but at a \( -45^\circ \) angle.
3. **Graph the circle \( x^2 + y^2 = 2 \)**: This is a circle centered at the origin with a radius of \( \sqrt{2} \).
### Step 3: Identify the feasible region
The feasible region is where all three conditions overlap:
- The area below the line \( y = x \) (including the line).
- The area above the line \( y = -x \) (including the line).
- The area inside the circle \( x^2 + y^2 = 2 \).
### Step 4: Determine the intersection points
To find the intersection points of the lines with the circle, we solve the equations:
1. For \( y = x \):
\[
x^2 + x^2 = 2 \implies 2x^2 = 2 \implies x^2 = 1 \implies x = 1 \text{ or } x = -1
\]
Thus, the points are \( (1, 1) \) and \( (-1, -1) \).
2. For \( y = -x \):
\[
x^2 + (-x)^2 = 2 \implies 2x^2 = 2 \implies x^2 = 1 \implies x = 1 \text{ or } x = -1
\]
Thus, the points are \( (1, -1) \) and \( (-1, 1) \).
### Step 5: Identify the vertices of the feasible region
The vertices of the feasible region defined by the intersection of the lines and the circle are:
- \( (1, 1) \)
- \( (1, -1) \)
- \( (-1, 1) \)
- \( (-1, -1) \)
### Step 6: Calculate the distances between the vertices
To find the diameter, we need to calculate the distances between all pairs of vertices and find the maximum distance.
1. Distance between \( (1, 1) \) and \( (1, -1) \):
\[
d = \sqrt{(1 - 1)^2 + (1 - (-1))^2} = \sqrt{0 + 4} = 2
\]
2. Distance between \( (1, 1) \) and \( (-1, 1) \):
\[
d = \sqrt{(1 - (-1))^2 + (1 - 1)^2} = \sqrt{4 + 0} = 2
\]
3. Distance between \( (1, -1) \) and \( (-1, -1) \):
\[
d = \sqrt{(1 - (-1))^2 + (-1 - (-1))^2} = \sqrt{4 + 0} = 2
\]
4. Distance between \( (-1, 1) \) and \( (-1, -1) \):
\[
d = \sqrt{(-1 - (-1))^2 + (1 - (-1))^2} = \sqrt{0 + 4} = 2
\]
5. Distance between \( (1, 1) \) and \( (-1, -1) \):
\[
d = \sqrt{(1 - (-1))^2 + (1 - (-1))^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2}
\]
6. Distance between \( (1, -1) \) and \( (-1, 1) \):
\[
d = \sqrt{(1 - (-1))^2 + (-1 - 1)^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2}
\]
### Step 7: Determine the maximum distance
The maximum distance calculated is \( 2\sqrt{2} \). However, since we are looking for the maximum distance between points in the feasible region, we must check the pairs that lie within the constraints. The maximum distance that satisfies all constraints is indeed \( 2 \).
### Conclusion
Thus, the diameter of the set \( S \) is \( 2 \).
