Proving Set Equality: A ∪ B — C A — C ∪ B — C

The problem of proving that the set equality ( A cup B — C A — C cup B — C ) holds for all sets ( A ), ( B ), and ( C ) is a classic exercise in set theory. This equality is particularly interesting because it involves both the union and set difference operations, and requires a systematic approach to verify it. We will delve into the proof by showing that each side of the equation is a subset of the other.

Introduction

Set theory is a fundamental branch of mathematics that deals with collections of objects. This article focuses on proving a specific set equality, which is a key concept for understanding more complex relationships between sets. The proof involves logical steps and set operations, providing a clear example of how set theory is applied in real-world scenarios.

Step 1: Prove ( A cup B — C subseteq A — C cup B — C )

To prove that ( A cup B — C subseteq A — C cup B — C ), we start by assuming that an element ( x ) is in ( A cup B — C ). By definition, this means that ( x ) is in ( A cup B ), and ( x ) is not in ( C ).

Given ( x in A cup B — C ), we know that ( x in A cup B ) and ( x otin C ).

By the definition of union, ( x in A cup B ) implies that ( x in A ) or ( x in B ) (or both).

Now we consider the two cases:

Case 1: If ( x in A ), since ( x otin C ), it follows that ( x in A — C ).

Case 2: If ( x in B ), similarly, since ( x otin C ), it follows that ( x in B — C ).

In either case, we conclude that ( x in A — C cup B — C ).

Step 2: Prove ( A — C cup B — C subseteq A cup B — C )

To prove that ( A — C cup B — C subseteq A cup B — C ), we assume that an element ( x ) is in ( A — C cup B — C ). By definition, this means that ( x ) is in ( A — C ) or ( x ) is in ( B — C ).

Given ( x in A — C cup B — C ), we know that ( x in A — C ) or ( x in B — C ).

By the definition of set difference, if ( x in A — C ), then ( x in A ) and ( x otin C ).

Similarly, if ( x in B — C ), then ( x in B ) and ( x otin C ).

Since ( x in A ) or ( x in B ) (or both) and ( x otin C ), it follows that ( x in A cup B ) and ( x otin C ).

By the definition of set difference, ( x in A cup B — C ).

Conclusion

Since we have shown both inclusions, we conclude that:

( A cup B — C subseteq A — C cup B — C )

( A — C cup B — C subseteq A cup B — C )

We can now conclude:

( A cup B — C A — C cup B — C )

Further Insights and Applications

The proof technique used here is a common approach in set theory and discrete mathematics. Understanding such proofs is essential for anyone working with data sets, databases, or any field where set operations are applied. The concept of set equality and subset relations are fundamental in various areas, including computer science, logic, and probability theory.

For example, in computer science, this proof might be used to optimize database queries or to ensure that certain conditions on data sets are met. In logic, it provides a clear example of how logical equivalences can be shown through set theory, and in probability, it can help in understanding joint and conditional probabilities.

Conclusion

Proving set equalities like ( A cup B — C A — C cup B — C ) is a powerful exercise that enhances logical thinking and problem-solving skills. Understanding these proofs is crucial for many advanced topics in mathematics and related fields. The techniques used here are not only applicable to set theory but also extend to a wide range of mathematical and practical problems.