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### 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]



Answer :

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]

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