Problem Statement:

We are asked to prove that if $A$ is a proper subset of $B$, then $C - B$ is a proper subset of $C - A$ for any sets $A$, $B$, and $C$.

Definitions:

1. Proper Subset: Set $A$ is a proper subset of $B$ (denoted $A \subsetneq B$) if:

   • $A \subseteq B$ (i.e., every element of $A$ is also in $B$),
   • and $A \neq B$ (i.e., there exists at least one element in $B$ that is not in $A$).

2. Set Difference: The set difference $C - X$ is defined as the set of elements in $C$ that are not in $X$. Formally, $C - X = \{ x \in C \mid x \notin X \}$.

Goal:

To prove that: $A \subsetneq B \implies C - B \subsetneq C - A$ which means:

• $C - B \subseteq C - A$ (subset condition),
• $C - B \neq C - A$ (proper subset condition).

Proof:

1. Subset Condition: $C - B \subseteq C - A$

Let $x \in C - B$. By definition of set difference, this means: $x \in C \quad \text{and} \quad x \notin B.$ Since $A \subseteq B$, we know that every element of $A$ is also in $B$, so: $x \notin B \implies x \notin A.$ Thus, $x \in C - A$. Therefore, we have shown that: $x \in C - B \implies x \in C - A,$ which proves that: $C - B \subseteq C - A.$

2. Proper Subset Condition: $C - B \neq C - A$

Since $A \subsetneq B$, there exists some element $y \in B$ such that $y \notin A$. Now, consider the case where $y \in C$. Since $y \in B$ and $y \notin A$, we have: $y \notin C - B \quad \text{(because } y \in B\text{)}.$ However, since $y \notin A$ and $y \in C$, we have: $y \in C - A.$ Thus, there exists an element $y \in C - A$ that is not in $C - B$, meaning $C - B \neq C - A$.

Conclusion:

We have shown both that:

1. $C - B \subseteq C - A$,
2. $C - B \neq C - A$.

Therefore, $C - B$ is a proper subset of $C - A$, and the statement is proved.

Do you have any questions or would you like further details on any part of this proof?

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Here are 5 related questions you might want to explore:

1. What is the difference between a subset and a proper subset?
2. Can we generalize this result to infinite sets?
3. How does this result change if $A = B$?
4. What is the relationship between set difference and set union?
5. How does the result change if $A$ is not a subset of $B$?

Tip: Always double-check the definitions of key terms like "subset" and "proper subset" when working with proofs involving sets. Misunderstandings in definitions can lead to incorrect conclusions.