Proving the Subset Relation: A-C is a Subset of B Only if A-B is a Subset of C

In this article, we delve into a rigorous mathematical proof to demonstrate the subset relation between sets. Specifically, we will explore the conditions under which the statement A - C ? B implies A - B ? C. We begin by defining the necessary concepts and then proceed with a detailed proof.

Definitions

To properly understand and prove the statement, it is essential to first define the key terms involved:

Set Difference

The set difference A - B is defined as the set of elements that are present in set A but not in set B. Formally, we write:

$$A - B { x in A mid x otin B }$$

Subset

We say that a set X is a subset of a set Y, denoted X subseteq Y, if every element of X is also an element of Y.

Proof

To prove the statement, we will proceed with an assumed condition and derive the conclusion step-by-step:

Assumption

Let us assume that A - C subseteq B. This implies that every element in A - C is also in B.

Take an Element

Consider any element y in A - B. By the definition of set difference, this means:

y in A y otin B

Consider the Implications

Since y in A and y otin B, we need to show that y in C. If y were not in C, then y would belong to A - C, as it is a member of A and not in C.

Contradiction

If y in A - C, then by our initial assumption, y must also be in B. However, this directly contradicts the fact that y otin B. This contradiction implies that our assumption that y otin C must be false, hence y in C.

Conclusion

Since our choice of y was arbitrary, we can conclude that A - B subseteq C. Therefore, the statement is proven.

Additional Insights

Additionally, we can explore the concept of proper subsets and how they relate to the original proof:

Proper Subset

A set X is a proper subset of a set Y if every element of X is an element of Y, but not every element of Y is an element of X. Formally, this is expressed as:

$$forall x in X, x in Y exists y in Y, y otin X$$

In this context, if A is a proper subset of B and B is a proper subset of C, we need to prove that the statements hold for A and C.

Proof for Proper Subset

Consider any element x in A. Since A is a proper subset of B, we know that x in B. And since B is a proper subset of C, we know that x in C. Therefore, we have:

$$forall x in A, x in C$$

The second part of the proof is slightly more complex but follows a similar logic. Since B is a proper subset of C, there exists at least one element y in C that is not in B. Given that all elements of A are in B, it follows that y otin A. Thus, we have:

$$exists y in C, y otin A$$

Depending on the level of formality required, the proof can be embellished with more precise language. However, these are the fundamental logical steps involved.

Conclusion

We have demonstrated through a structured and formal proof that the subset relation A - C ? B implies A - B ? C. This proof not only validates the original statement but also provides insight into the interplay between set differences and subset relations.