Question

transformations test (topic 4)
let k and y be the following sets:
k = rigid motion transformations
y = all transformations
what is the set for the following?
k ∩ y

options:

• transformations, rotations, reflections, and dilations
• dilations
• translations
• rotations and reflections

Explanation:

Step1: Understand Set Definitions

- \( X \) is the set of **Rigid Motion Transformations** (translations, rotations, reflections—these preserve distance and shape).
- \( Y \) is the set of **All Transformations** (includes rigid motions and non - rigid motions like dilations, which change size).
- \( X' \) (complement of \( X \)) is the set of transformations that are **not** rigid motions (i.e., non - rigid motions, like dilations).

Step2: Analyze \( X' \cap Y \)

- The intersection \( A\cap B \) is the set of elements common to both \( A \) and \( B \).
- \( Y \) contains all transformations, and \( X' \) contains non - rigid motions. So \( X' \cap Y \) is the set of non - rigid motions (since non - rigid motions are in \( Y \) and in \( X' \)). But wait, let's re - evaluate the options. Wait, maybe a misinterpretation: Wait, rigid motions are translations, rotations, reflections. So \( X=\{\text{translations, rotations, reflections}\} \), \( Y = \{\text{all transformations: translations, rotations, reflections, dilations, etc.}\} \). Then \( X' \) is the set of transformations not in \( X \), so \( X'=\{\text{dilations, etc. (non - rigid)}\} \). But the options: Wait, maybe the question has a typo or my initial understanding is wrong. Wait, maybe \( X \) is "Translation Transformations" (only translations), then \( X' \) is all transformations except translations. \( Y \) is all transformations. Then \( X' \cap Y \) is all transformations except translations, which would be rotations, reflections, dilations. But looking at the options, the first option is "Translations, Rotations, Reflections, and Dilations"—no, that can't be. Wait, maybe \( X \) is "Rigid Motion Transformations" (translations, rotations, reflections) and \( Y \) is all transformations. Then \( X' \) is non - rigid (dilations). But the option "Rotations and Reflections"—no. Wait, maybe the original problem: Let's re - read. The user's image: \( X=\{\text{Rigid Motion Transformations}\} \) (translations, rotations, reflections), \( Y = \{\text{All Transformations}\} \) (includes rigid and non - rigid like dilations). Then \( X' \) is the complement of \( X \) in the universal set (which we can assume is \( Y \) here, since \( Y \) is all transformations). Wait, if the universal set is \( Y \), then \( X'=Y - X \), which is non - rigid transformations (dilations). But the options: Wait, the first option is "Translations, Rotations, Reflections, and Dilations"—no. Wait, maybe the question is \( X=\{\text{Translation Transformations}\} \), so \( X' \) is all transformations except translations. Then \( X' \cap Y \) is all transformations except translations, which are rotations, reflections, dilations. But the option "Rotations and Reflections"—no. Wait, maybe the correct approach is:

Rigid motions (X) are translations, rotations, reflections. All transformations (Y) include rigid motions and non - rigid (dilations). \( X' \) is non - rigid (dilations). But the option "Rotations and Reflections"—no. Wait, maybe the question has \( X \) as "Translations" (only translations), so \( X' \) is rotations, reflections, dilations. Then \( X' \cap Y \) is rotations, reflections, dilations? No, the first option is "Translations, Rotations, Reflections, and Dilations"—that would be \( Y \). Wait, I think I made a mistake. Let's recall set theory: \( X' \) is the complement of \( X \), so if the universal set is the set of all transformations (Y), then \( X'=Y - X \). So if \( X \) is rigid motions (translations, rotations, reflections), then \( X' \) is non - rigid motions (dilations). But the options: Wait, the option "Rotations and Reflections"—no. Wait, maybe the original problem's \( X \) is "Translations", so \( X \) is just translations. Then \( X' \) is all transformations except translations (rotations, reflections, dilations). \( Y \) is all transformations. So \( X' \cap Y = X' \), which is rotations, reflections, dilations. But the first option is "Translations, Rotations, Reflections, and Dilations"—that's \( Y \). The option "Rotations and Reflections"—no, because dilations are also non - translation. Wait, maybe the correct answer is the first option? No, that can't be. Wait, maybe the question is \( X=\{\text{Translations}\} \), \( Y=\{\text{All Transformations}\} \), then \( X' \cap Y \) is all transformations except translations, which are rotations, reflections, dilations. But the first option includes translations, which is wrong. Wait, maybe the problem has a different definition. Alternatively, maybe \( X \) is "Rigid Motions" (translations, rotations, reflections) and \( Y \) is "All Transformations", then \( X' \cap Y \) is non - rigid motions (dilations), but there's no option for dilations alone. Wait, the option "Rotations and Reflections"—no, they are in \( X \). Wait, I think I messed up. Let's look at the options:

Option 1: Translations, Rotations, Reflections, and Dilations (this is \( Y \), all transformations)

Option 2: Dilations (non - rigid)

Option 3: Translations (in \( X \), so not in \( X' \))

Option 4: Rotations and Reflections (in \( X \), so not in \( X' \))

Wait, if \( X \) is rigid motions (translations, rotations, reflections), then \( X' \) is non - rigid (dilations). So \( X' \cap Y \) is dilations (Option 2). But maybe the original problem's \( X \) is "Translations", so \( X' \) is rotations, reflections, dilations. Then \( X' \cap Y \) is rotations, reflections, dilations. But the first option includes translations, which is wrong. Alternatively, maybe the question has a typo and \( X \) is "Translations", \( Y \) is "All Transformations", then \( X' \cap Y \) is rotations, reflections, dilations, but the first option is wrong. Wait, maybe the correct answer is Option 2: Dilations. But let's re - check.

Wait, the problem says \( X = \{\text{Rigid Motion Transformations}\} \) (translations, rotations, reflections), \( Y=\{\text{All Transformations}\} \) (includes rigid and non - rigid like dilations). Then \( X' \) (complement of \( X \)) is the set of transformations not in \( X \), so non - rigid (dilations). Then \( X' \cap Y \) is the set of elements in both \( X' \) and \( Y \), which is \( X' \) (since \( X' \subseteq Y \)). So \( X' \cap Y=X'=\{\text{dilations}\} \). So the answer should be the option with Dilations (Option 2: Dilations). But let's check the options again:

- Option 1: Translations, Rotations, Reflections, and Dilations (this is \( Y \), all transformations, so not \( X' \cap Y \))
- Option 2: Dilations (this is \( X' \cap Y \), since \( X' \) is non - rigid, and \( Y \) includes dilations)
- Option 3: Translations (in \( X \), so not in \( X' \))
- Option 4: Rotations and Reflections (in \( X \), so not in \( X' \))

So the correct answer is Option 2: Dilations. Wait, but the user's image shows a blue dot on the first option, but that might be a mistake. Let's go with set theory:

\( X \): Rigid Motions (translations, rotations, reflections)

\( Y \): All Transformations (translations, rotations, reflections, dilations)

\( X' \): Transformations not in \( X \) → dilations

\( X' \cap Y \): Elements in both \( X' \) and \( Y \) → dilations (since \( X' \subseteq Y \))

So the answer is the option with Dilations.

Answer:

B. Dilations