Answered

Westonci.ca is the premier destination for reliable answers to your questions, provided by a community of experts. Discover comprehensive answers to your questions from knowledgeable professionals on our user-friendly platform. Our platform provides a seamless experience for finding reliable answers from a network of experienced professionals.

### Definition of Proper Subset

If every element of set [tex]\( A \)[/tex] is also an element of set [tex]\( B \)[/tex], but set [tex]\( B \)[/tex] has more elements than set [tex]\( A \)[/tex], then [tex]\( A \)[/tex] is a proper subset of [tex]\( B \)[/tex].

Notation: [tex]\( A \subset B \)[/tex]

#### Example 9
Let [tex]\( A = \{1, 2, 3, 4\} \)[/tex], [tex]\( B = \{x \mid x \in \mathbb{Z}\} \)[/tex], and [tex]\( C = \{x \mid x \text{ is a factor of 12}\} \)[/tex].

Fill in the blanks with the correct symbols ([tex]\(\subset, \not\subset, =\)[/tex]).

a. [tex]\( A \quad \underline{\qquad} \quad B \)[/tex]

b. [tex]\( A \quad \underline{\qquad} \quad C \)[/tex]

c. [tex]\( A \quad \underline{\qquad} \quad A \)[/tex]

d. [tex]\( B \quad \underline{\qquad} \quad A \)[/tex]

e. [tex]\( B \quad \underline{\qquad} \quad C \)[/tex]

f. [tex]\( C \quad \underline{\qquad} \quad A \)[/tex]

g. [tex]\( C \quad \underline{\qquad} \quad B \)[/tex]

h. [tex]\( C \quad \underline{\qquad} \quad C \)[/tex]

Sagot :

GinnyAnswer
Given the sets [tex]\( A = \{1, 2, 3, 4\} \)[/tex], [tex]\( B = \{x \mid x \in \mathbb{Z}\} \)[/tex] (meaning [tex]\( B \)[/tex] represents all integers), and [tex]\( C = \{1, 2, 3, 4, 6, 12\} \)[/tex] (the set of factors of 12), we need to determine the relationships between these sets using the symbols [tex]\( \subset \)[/tex], [tex]\( \not\subset \)[/tex], and [tex]\( = \)[/tex].

### Solution:

1. a. [tex]\( A \qquad B \)[/tex]

Explanation: Every element of [tex]\( A \)[/tex] is an integer, and [tex]\( B \)[/tex] consists of all integers. Since [tex]\( A \)[/tex] is a subset of [tex]\( B \)[/tex] and [tex]\( B \)[/tex] has more elements than [tex]\( A \)[/tex], [tex]\( A \)[/tex] is a proper subset of [tex]\( B \)[/tex].

Answer: [tex]\( A \subset B \)[/tex]

2. b. [tex]\( A \qquad C \)[/tex]

Explanation: Every element of [tex]\( A \)[/tex] (which are 1, 2, 3, and 4) is in [tex]\( C \)[/tex]. While [tex]\( A \)[/tex] is a proper subset of [tex]\( C \)[/tex] because [tex]\( C \)[/tex] has additional elements (6 and 12), [tex]\( A \subset C \)[/tex].

Answer: [tex]\( A \subset C \)[/tex]

3. c. [tex]\( A \qquad A \)[/tex]

Explanation: Since [tex]\( A \)[/tex] is identical to itself, it is neither a proper subset nor not a subset of itself. It is equal to [tex]\( A \)[/tex].

Answer: [tex]\( A = A \)[/tex]

4. d. [tex]\( B \qquad A \)[/tex]

Explanation: Since [tex]\( B \)[/tex] contains all integers, it is much larger than [tex]\( A \)[/tex]. Therefore, [tex]\( B \)[/tex] is not a subset of [tex]\( A \)[/tex].

Answer: [tex]\( B \not\subset A \)[/tex]

5. e. [tex]\( B \qquad C \)[/tex]

Explanation: Again, [tex]\( B \)[/tex] includes all possible integers, far exceeding the elements in [tex]\( C \)[/tex], so [tex]\( B \)[/tex] is not a subset of [tex]\( C \)[/tex].

Answer: [tex]\( B \not\subset C \)[/tex]

6. f. [tex]\( C \qquad A \)[/tex]

Explanation: Set [tex]\( C \)[/tex] contains the elements 1, 2, 3, 4, 6, and 12, while [tex]\( A \)[/tex] contains only 1, 2, 3, and 4. Therefore, some elements of [tex]\( C \)[/tex] are not in [tex]\( A \)[/tex], indicating [tex]\( C \)[/tex] is not a subset of [tex]\( A \)[/tex].

Answer: [tex]\( C \not\subset A \)[/tex]

7. g. [tex]\( C \qquad B \)[/tex]

Explanation: Every element of [tex]\( C \)[/tex] is an integer, and thus is in [tex]\( B \)[/tex]. Since [tex]\( B \)[/tex] contains more integers than just what is in [tex]\( C \)[/tex], [tex]\( C \)[/tex] is a proper subset of [tex]\( B \)[/tex].

Answer: [tex]\( C \subset B \)[/tex]

8. h. [tex]\( C \qquad C \)[/tex]

Explanation: Since [tex]\( C \)[/tex] is equal to itself, it is neither a proper subset nor not a subset. It is equal.

Answer: [tex]\( C = C \)[/tex]

### Summary:
a. [tex]\( A \subset B \)[/tex]
b. [tex]\( A \subset C \)[/tex]
c. [tex]\( A = A \)[/tex]
d. [tex]\( B \not\subset A \)[/tex]
e. [tex]\( B \not\subset C \)[/tex]
f. [tex]\( C \not\subset A \)[/tex]
g. [tex]\( C \subset B \)[/tex]
h. [tex]\( C = C \)[/tex]
Thank you for choosing our service. We're dedicated to providing the best answers for all your questions. Visit us again. Your visit means a lot to us. Don't hesitate to return for more reliable answers to any questions you may have. We're glad you chose Westonci.ca. Revisit us for updated answers from our knowledgeable team.