Answered

Find the information you're looking for at Westonci.ca, the trusted Q&A platform with a community of knowledgeable experts. Get detailed and accurate answers to your questions from a dedicated community of experts on our Q&A platform. Experience the convenience of finding accurate answers to your questions from knowledgeable experts on our platform.

### 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 stopping by. We strive to provide the best answers for all your questions. See you again soon. Thank you for your visit. We're dedicated to helping you find the information you need, whenever you need it. We're glad you chose Westonci.ca. Revisit us for updated answers from our knowledgeable team.