Let's analyze the sets $A = \{1\}$, $B = (0, 1)$, and $C = [0, 1]$ to determine which sets are subsets of which.

Definitions:

• $A = \{1\}$: This is a set containing a single element, the number 1.
• $B = (0, 1)$: This represents an open interval, meaning it contains all real numbers strictly between 0 and 1, but does not include 0 or 1.
• $C = [0, 1]$: This represents a closed interval, meaning it contains all real numbers between 0 and 1, including both 0 and 1.

(a) Which sets are subsets of which?

1. Is $A \subseteq B$?

   • The set $A$ contains the element 1. However, $B$ is an open interval and does not include 1, so $A$ is not a subset of $B$.

2. Is $A \subseteq C$?

   • The set $A$ contains the element 1, and $C$ includes 1 as part of the closed interval. Therefore, $A$ is a subset of $C$.

3. Is $B \subseteq A$?

   • The set $B$ contains an infinite number of points between 0 and 1, none of which are included in $A$, so $B$ is not a subset of $A$.

4. Is $B \subseteq C$?

   • The set $B$ contains all points strictly between 0 and 1, and all these points are also included in $C$. Hence, $B$ is a subset of $C$.

5. Is $C \subseteq A$?

   • The set $C$ contains all points between 0 and 1, including 0 and 1. Since $A$ only contains the point 1, $C$ is not a subset of $A$.

6. Is $C \subseteq B$?

   • The set $C$ includes the endpoints 0 and 1, which are not in $B$. Therefore, $C$ is not a subset of $B$.

Summary:

• $A \subseteq C$
• $B \subseteq C$
• None of the other pairings are subsets.

Would you like any further details or explanations?

Related Questions:

1. What is the difference between an open interval and a closed interval?
2. Can you explain how to determine if one set is a subset of another?
3. What are some real-life examples where open and closed intervals are used?
4. How does the concept of subsets extend to higher-dimensional spaces?
5. What happens if we consider the union or intersection of these sets $A$, $B$, and $C$?
6. How do the properties of open and closed intervals relate to limits in calculus?
7. What is the complement of each of these sets in the real number system?
8. How would the analysis change if we considered $A = \{0, 1\}$?

Tip: When dealing with subsets, it's essential to clearly understand the elements contained within each set and whether those elements are included in the other set.