To determine which set is a subset of the set [tex]\( I \)[/tex], we need to analyze the definitions and properties of the given sets:
1. Definitions of 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]
2. Properties of Triangles:
- An equilateral triangle is also an isosceles triangle because it has at least two equal sides.
- A scalene triangle has no equal sides and thus is not an isosceles triangle.
- An acute triangle has all its angles less than 90 degrees, but this does not give any information about the lengths of its sides, so it could be equilateral, isosceles, or scalene.
3. Checking Subset Relations:
- Equilateral triangles (E): As noted, all equilateral triangles are isosceles because they have at least two equal sides. Therefore, [tex]\( E \subseteq I \)[/tex].
- Scalene triangles (S): By definition, a scalene triangle has all sides of different lengths, so it cannot be isosceles. Therefore, [tex]\( S \not\subseteq I \)[/tex].
- Acute triangles (A): Acute triangles can be equilateral, isosceles, or scalene. Therefore, not all acute triangles are isosceles, and we cannot conclude [tex]\( A \subseteq I \)[/tex].
Given this analysis, among the given options [tex]\( E, S, \)[/tex] and [tex]\( A \)[/tex]:
The only set that is a subset of [tex]\( I \)[/tex] is [tex]\( E \)[/tex].
Therefore, the correct answer is:
[tex]\[ \boxed{E} \][/tex]
