Let's analyze the given sets and determine which ones are subsets of [tex]\( I \)[/tex] (the set of isosceles triangles).
We have the following sets:
- [tex]\( U = \{ \text{all triangles} \} \)[/tex]
- [tex]\( E = \{ x \mid x \in U \text{ and } x \text{ is equilateral} \} \)[/tex]
- [tex]\( I = \{ x \mid x \in U \text{ and } x \text{ is isosceles} \} \)[/tex]
- [tex]\( S = \{ x \mid x \in U \text{ and } x \text{ is scalene} \} \)[/tex]
- [tex]\( A = \{ x \mid x \in U \text{ and } x \text{ is acute} \} \)[/tex]
- [tex]\( O = \{ x \mid x \in U \text{ and } x \text{ is obtuse} \} \)[/tex]
- [tex]\( R = \{ x \mid x \in U \text{ and } x \text{ is right} \} \)[/tex]
We need to determine if any of [tex]\( E \)[/tex], [tex]\( S \)[/tex], [tex]\( A \)[/tex], or [tex]\( R \)[/tex] are subsets of [tex]\( I \)[/tex].
1. Equilateral Triangles ([tex]\( E \)[/tex]):
- An equilateral triangle has all three sides equal.
- By definition, an isosceles triangle has at least two sides equal.
- Since equilateral triangles also have at least two sides equal, every equilateral triangle is also an isosceles triangle.
- Therefore, [tex]\( E \subseteq I \)[/tex].
2. Scalene Triangles ([tex]\( S \)[/tex]):
- A scalene triangle has all three sides of different lengths.
- An isosceles triangle, on the other hand, must have at least two sides equal.
- Therefore, a scalene triangle cannot be an isosceles triangle.
- Thus, [tex]\( S \not\subseteq I \)[/tex].
3. Acute Triangles ([tex]\( A \)[/tex]):
- An acute triangle has all three interior angles less than 90 degrees.
- An acute triangle can have all sides equal (making it an equilateral triangle and thereby an isosceles triangle), or two sides equal (making it directly an isosceles triangle), or no sides equal (not an isosceles triangle).
- Therefore, not all acute triangles are isosceles.
- Hence, [tex]\( A \not\subseteq I \)[/tex].
4. Right Triangles ([tex]\( R \)[/tex]):
- A right triangle has one interior angle equal to 90 degrees.
- A right triangle can be isosceles if the two legs (sides opposite the right angle) are of equal length.
- However, not all right triangles are isosceles; some may have three different side lengths.
- Therefore, [tex]\( R \not\subseteq I \)[/tex].
Based on this analysis, the sets [tex]\( E \)[/tex] (equilateral triangles) is the subset of [tex]\( I \)[/tex] (isosceles triangles).
Thus, the answer is:
[tex]\[ E \][/tex]
