To determine which set is a subset of [tex]\(I\)[/tex] (isosceles triangles), let's analyze the definitions of the given sets and their relationships.
Given sets:
- [tex]\(U =\{\)[/tex] all triangles [tex]\(\}\)[/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]
Now let's look at the specific properties of these sets:
1. E (Equilateral triangles):
- An equilateral triangle has all three sides of equal length.
- This means that any equilateral triangle is also isosceles, as having all sides equal satisfies the condition of having at least two sides equal.
- Therefore, [tex]\(E \subseteq I\)[/tex].
2. S (Scalene triangles):
- A scalene triangle has all sides of different lengths.
- Since no sides are equal, scalene triangles cannot be isosceles.
- Thus, [tex]\(S \not\subseteq I\)[/tex].
3. A (Acute triangles):
- An acute triangle has all angles less than 90 degrees.
- Acute triangles can have sides of different lengths (scalene) or at least two sides equal (isosceles), or even all sides equal (equilateral which is a special case of isosceles).
- Therefore, acute triangles might intersect with isosceles but are not entirely contained within isosceles.
- Thus, [tex]\(A \not\subseteq I\)[/tex].
4. R (Right triangles):
- A right triangle has one angle exactly 90 degrees.
- Right triangles can be scalene (if sides are all different) or isosceles (if two sides are equal).
- However, right triangles as a whole are not necessarily isosceles.
- Thus, [tex]\(R \not\subseteq I\)[/tex].
Based on the properties and definitions above, the correct subset of [tex]\(I\)[/tex] is:
E (Equilateral triangles).
So, [tex]\(E\)[/tex] is a subset of [tex]\(I\)[/tex], and the correct answer is:
[tex]\[ \boxed{E} \][/tex]
