Answered

At Westonci.ca, we make it easy for you to get the answers you need from a community of knowledgeable individuals. Join our platform to connect with experts ready to provide precise answers to your questions in various areas. Connect with a community of professionals ready to help you find accurate solutions to your questions quickly and efficiently.

### 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]
Thanks for using our platform. We're always here to provide accurate and up-to-date answers to all your queries. We hope this was helpful. Please come back whenever you need more information or answers to your queries. Thank you for visiting Westonci.ca. Stay informed by coming back for more detailed answers.