Question

select the correct answer.
given
d = {x | x is a whole number}
e = {x | x is a perfect square between 49 and 100}
f = {x | x is an even number between 10 and 20}
the expression d ∪ f means
the expression d ∩ f means
the expression d ∩ e means
the expression e ∩ f means
the expression d ∩ (e ∪ f) means

Explanation:

To solve this, we first define the sets:

• \( D = \{x \mid x \text{ is a whole number}\} = \{0, 1, 2, 3, \dots\} \)
• \( E = \{x \mid x \text{ is a perfect square between } 49 \text{ and } 100\} \). Perfect squares: \( 8^2 = 64 \), \( 9^2 = 81 \), so \( E = \{64, 81\} \).
• \( F = \{x \mid x \text{ is an even number between } 10 \text{ and } 20\} \). Even numbers: \( 12, 14, 16, 18 \), so \( F = \{12, 14, 16, 18\} \).

1. \( D \cup F \)

The union (\( \cup \)) of two sets includes all elements in either set. Since \( F \)’s elements are whole numbers (and \( D \) is all whole numbers), \( D \cup F = D \) (all whole numbers, as \( F \) is a subset of \( D \)).

2. \( D \cap F \)

The intersection (\( \cap \)) includes elements common to both sets. \( F \)’s elements are whole numbers, so \( D \cap F = F = \{12, 14, 16, 18\} \).

3. \( D \cap E \)

Intersection of \( D \) (whole numbers) and \( E \) (perfect squares 64, 81). Since \( E \)’s elements are whole numbers, \( D \cap E = E = \{64, 81\} \).

4. \( E \cap F \)

Intersection of \( E = \{64, 81\} \) and \( F = \{12, 14, 16, 18\} \). No common elements, so \( E \cap F = \emptyset \) (empty set).

5. \( D \cap (E \cup F) \)

First, find \( E \cup F \): all elements in \( E \) or \( F \), so \( E \cup F = \{12, 14, 16, 18, 64, 81\} \). Then, intersect with \( D \) (all whole numbers). Since \( E \cup F \)’s elements are whole numbers, \( D \cap (E \cup F) = E \cup F = \{12, 14, 16, 18, 64, 81\} \).

Final Answers (for each dropdown, select the description matching the set operation result):

• \( D \cup F \): All whole numbers (since \( F \subseteq D \))
• \( D \cap F \): Even numbers between 10 and 20 (i.e., \( F \))
• \( D \cap E \): Perfect squares between 49 and 100 (i.e., \( E \))
• \( E \cap F \): No elements (empty set)
• \( D \cap (E \cup F) \): Elements in \( E \) or \( F \) (i.e., \( E \cup F \))

(Note: The exact dropdown options were not provided, but the above describes the correct interpretation of each set operation.)

Answer:

To solve this, we first define the sets:

• \( D = \{x \mid x \text{ is a whole number}\} = \{0, 1, 2, 3, \dots\} \)
• \( E = \{x \mid x \text{ is a perfect square between } 49 \text{ and } 100\} \). Perfect squares: \( 8^2 = 64 \), \( 9^2 = 81 \), so \( E = \{64, 81\} \).
• \( F = \{x \mid x \text{ is an even number between } 10 \text{ and } 20\} \). Even numbers: \( 12, 14, 16, 18 \), so \( F = \{12, 14, 16, 18\} \).

1. \( D \cup F \)

The union (\( \cup \)) of two sets includes all elements in either set. Since \( F \)’s elements are whole numbers (and \( D \) is all whole numbers), \( D \cup F = D \) (all whole numbers, as \( F \) is a subset of \( D \)).

2. \( D \cap F \)

The intersection (\( \cap \)) includes elements common to both sets. \( F \)’s elements are whole numbers, so \( D \cap F = F = \{12, 14, 16, 18\} \).

3. \( D \cap E \)

Intersection of \( D \) (whole numbers) and \( E \) (perfect squares 64, 81). Since \( E \)’s elements are whole numbers, \( D \cap E = E = \{64, 81\} \).

4. \( E \cap F \)

Intersection of \( E = \{64, 81\} \) and \( F = \{12, 14, 16, 18\} \). No common elements, so \( E \cap F = \emptyset \) (empty set).

5. \( D \cap (E \cup F) \)

First, find \( E \cup F \): all elements in \( E \) or \( F \), so \( E \cup F = \{12, 14, 16, 18, 64, 81\} \). Then, intersect with \( D \) (all whole numbers). Since \( E \cup F \)’s elements are whole numbers, \( D \cap (E \cup F) = E \cup F = \{12, 14, 16, 18, 64, 81\} \).

Final Answers (for each dropdown, select the description matching the set operation result):

• \( D \cup F \): All whole numbers (since \( F \subseteq D \))
• \( D \cap F \): Even numbers between 10 and 20 (i.e., \( F \))
• \( D \cap E \): Perfect squares between 49 and 100 (i.e., \( E \))
• \( E \cap F \): No elements (empty set)
• \( D \cap (E \cup F) \): Elements in \( E \) or \( F \) (i.e., \( E \cup F \))

(Note: The exact dropdown options were not provided, but the above describes the correct interpretation of each set operation.)