Proving the Symmetric Difference using Set Operations

Introduction

In set theory, the symmetric difference of two sets is a fundamental concept that is often defined using set operations such as union and intersection. This article will explore how to prove that the symmetric difference A Δ B can be expressed as A ∪ B - A ∩ B. We will break down the proof step by step and discuss the meanings of the involved set operations.

Definitions

Before delving into the proof, it is important to define the key terms. The symmetric difference of two sets A and B, denoted as A Δ B, represents the set of elements that are present in either A or B but not in both. Mathematically, this can be expressed as:

A Δ B (A - B) ∪ (B - A)

On the other hand, the union of sets A and B, denoted as A ∪ B, is the set containing all elements that are in A, in B, or in both.

A ∪ B

The intersection of sets A and B, denoted as A ∩ B, is the set containing all elements that are in both A and B.

A ∩ B

Proof

To prove that A Δ B A ∪ B - A ∩ B, we need to show that both sides of the equation represent the same set. This can be done by proving that each side is a subset of the other. Let's proceed step by step.

Step 1: Show A Δ B ? A ∪ B - A ∩ B

Assume that x ∈ A Δ B. By the definition of symmetric difference, this means:

x ∈ A and x ? B or x ∈ B and x ? A

In either case, x ∈ A ∪ B (since x is in either A or B). Furthermore, x ? A ∩ B (since x cannot be in both A and B). Therefore, x ∈ A ∪ B - A ∩ B.

Step 2: Show A ∪ B - A ∩ B ? A Δ B

Assume that x ∈ A ∪ B - A ∩ B. This means:

x ∈ A ∪ B and x ? A ∩ B

From x ∈ A ∪ B, we have:

x ∈ A or x ∈ B

Since x ? A ∩ B, it cannot be in both sets. Therefore:

if x ∈ A, then x ? B, which means x ∈ A - B. if x ∈ B, then x ? A, which means x ∈ B - A.

In either case, x ∈ A Δ B.

Since we have shown both inclusions, we conclude that:

A Δ B A ∪ B - A ∩ B

Therefore, the symmetric difference of two sets can be expressed as the union of the sets minus their intersection.

Conclusion

The proof provided above demonstrates that the symmetric difference A Δ B is indeed equivalent to the set operation A ∪ B - A ∩ B. This equivalence is a fundamental concept in set theory and has numerous applications in various fields such as computer science and mathematics.