Consider the following sets:
[tex]R =\{x \mid x[/tex] is the set of rectangles [tex]\}[/tex]
[tex]P =\{x \mid x[/tex] is the set of parallelograms [tex]\}[/tex]
[tex]T =\{x \mid x[/tex] is the set of triangles [tex]\}[/tex]
[tex]I =\{x \mid x[/tex] is the set of isosceles triangles [tex]\}[/tex]
[tex]E =\{x \mid x[/tex] is the set of equilateral triangles [tex]\}[/tex]
[tex]S =\{x \mid x[/tex] is the set of scalene triangles [tex]\}[/tex]
Which statements are correct? Check all that apply.
A. T is a subset of P.
B. E is a subset of I.
C. S is a subset of T.
D. [tex]R \subset P[/tex]
Answer (2)
E ⊂ I , S ⊂ T , R ⊂ P
Explanation
Analyze each statement. Let's analyze each statement to determine its correctness.
T is a subset of P: This statement asks if all triangles are parallelograms. A parallelogram is a quadrilateral (a four-sided figure) with opposite sides parallel. A triangle is a three-sided figure. Therefore, a triangle cannot be a parallelogram. This statement is false.
E is a subset of I: This statement asks if all equilateral triangles are isosceles triangles. An equilateral triangle has three equal sides and three equal angles. An isosceles triangle has at least two equal sides and two equal angles. Since an equilateral triangle has three equal sides, it automatically has at least two equal sides. Therefore, every equilateral triangle is also an isosceles triangle. This statement is true.
S is a subset of T: This statement asks if all scalene triangles are triangles. A scalene triangle is a triangle with all three sides of different lengths. Therefore, a scalene triangle is, by definition, a triangle. This statement is true.
R is a subset of P: This statement asks if all rectangles are parallelograms. A rectangle is a quadrilateral with four right angles. A parallelogram is a quadrilateral with opposite sides parallel. In a rectangle, opposite sides are parallel, and opposite sides are equal. Therefore, every rectangle is also a parallelogram. This statement is true.
Determine correct statements. Based on the analysis above, the following statements are correct:
E is a subset of I
S is a subset of T
R is a subset of P
Final Answer. Therefore, the correct statements are:
E is a subset of I
S is a subset of T
R is a subset of P
Examples
Understanding set theory and subsets helps in various fields. For example, in computer science, you can define sets of data structures. All linked lists form a set, and all singly linked lists form a subset of that set. Similarly, in biology, all mammals form a set, and all primates form a subset of mammals. This hierarchical classification is fundamental to organizing and understanding complex systems.
The correct statements are that all equilateral triangles are isosceles triangles (E is a subset of I), all scalene triangles are triangles (S is a subset of T), and all rectangles are parallelograms (R is a subset of P). However, not all triangles are parallelograms (T is not a subset of P).
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