Demonstrate that a set of four subset elements forms a Klein four-group using symmetric difference operation | Step-by-Step Solution

What We're Solving:

We need to prove that when we take all possible subsets of a two-element set and use the symmetric difference operation (∆), we get a Klein four-group. This is a beautiful example of how familiar set operations can create important algebraic structures!

The Approach:

To prove something is a Klein four-group, we need to show it satisfies the group axioms AND has the special Klein four-group property. Here's our strategy:

• 1. Identify our set X and find all its subsets
• 2. Define our operation (symmetric difference ∆)
• 3. Verify the four group axioms: closure, associativity, identity, and inverses
• 4. Show it's specifically a Klein four-group: every non-identity element has order 2

Step-by-Step Solution:

Step 1: Set up our elements Let X = {a, b} (our two-element set) Then V = {∅, {a}, {b}, {a,b}} (all possible subsets of X)

Step 2: Understand symmetric difference (∆) For any two sets A and B: A ∆ B = (A ∪ B) - (A ∩ B) In other words, A ∆ B contains elements that are in exactly one of A or B, but not both.

Step 3: Verify closure We need to check that A ∆ B is always in V when A, B ∈ V. Create a operation table:

• ∅ ∆ ∅ = ∅ ✓
• ∅ ∆ {a} = {a} ✓
• {a} ∆ {b} = {a,b} ✓
• {a} ∆ {a,b} = {b} ✓

(Continue for all 16 combinations)

Step 4: Check associativity Symmetric difference is always associative for sets. You can verify this with the definition or use the fact that ∆ corresponds to addition in ℤ₂ × ℤ₂.

Step 5: Find the identity element The empty set ∅ is our identity because A ∆ ∅ = A for any set A.

Step 6: Verify inverses Each element is its own inverse!

• ∅ ∆ ∅ = ∅ ✓
• {a} ∆ {a} = ∅ ✓
• {b} ∆ {b} = ∅ ✓
• {a,b} ∆ {a,b} = ∅ ✓

Step 7: Confirm Klein four-group structure Every non-identity element has order 2 (is its own inverse), and we have exactly 4 elements. The group is abelian since symmetric difference is commutative.

The Answer:

Yes! V with the symmetric difference operation forms a Klein four-group (V₄). It satisfies all group axioms, has 4 elements, is abelian, and every non-identity element has order 2. This group is isomorphic to ℤ₂ × ℤ₂.

Memory Tip:

Remember "Klein = Clean slate twice"! In a Klein four-group, doing any operation twice with the same element gets you back to the identity (like erasing on a clean slate). Also, the symmetric difference visually shows you what's "different" between two sets - perfect for a group where everything undoes itself!

The beauty here is seeing how simple set operations create elegant algebraic structures. You've just connected basic set theory with abstract algebra! 🎉