Symmetric difference of two sets

Symmetric Difference of Two Sets Definition

The symmetric difference of two sets is a key concept in set theory. It refers to the set of elements that are present in either of the two sets, but not in both. In other words, it includes elements that are unique to each set and excludes those that are shared. This operation is widely used in mathematics, computer science, and data analysis for comparing data sets and finding exclusive elements.

Symmetric Difference of Two Sets Symbol

The symmetric difference of two sets A and B is denoted by the symbol Δ, written as A Δ B or sometimes as A ⊕ B or A △ B. This notation is standard across mathematical texts and is easily recognized in Venn diagrams and set operations.

Symmetric Difference of Two Sets Formula

There are two commonly used formulas for the symmetric difference of two sets A and B:

• A Δ B = (A - B) ∪ (B - A)
• A Δ B = (A ∪ B) - (A ∩ B)

Both formulas yield the same result: the set of elements that are in A or B, but not in both.

Symmetric Difference of Two Sets Venn Diagram

A symmetric difference of two sets Venn diagram visually represents the elements that belong exclusively to each set. In the diagram, the non-overlapping parts of circles A and B are shaded, indicating the elements that are in A or B, but not in their intersection.

Symmetric Difference of Two Sets Examples

Example 1:

Let A = {1, 2, 3, 4} and B = {3, 4, 5, 6}.

• A - B = {1, 2}
• B - A = {5, 6}
• A Δ B = {1, 2, 5, 6} (elements unique to each set)

Example 2:

Let A = {a, b, c} and B = {b, c, d, e}.

• A - B = {a}
• B - A = {d, e}
• A Δ B = {a, d, e}

Symmetric Difference of Two Sets A and B

For any two sets A and B, the symmetric difference of two sets A and B can be found using the formula and steps above. It is always denoted by A Δ B and includes elements that are in A or B, but not in both.

Symmetric Difference of Two Sets Proof

To prove the formula for symmetric difference, consider:

• (A - B) contains elements in A but not in B.
• (B - A) contains elements in B but not in A.
• Their union, (A - B) ∪ (B - A), thus contains all elements unique to A or B.
  Alternatively, (A ∪ B) - (A ∩ B) removes the intersection from the union, leaving only unique elements.

Symmetric Difference of Two Sets Calculator

A symmetric difference of two sets calculator is a digital tool where you input two sets, and it computes the symmetric difference for you. These calculators are widely available online and are helpful for checking your work or handling large data sets.

Properties of Symmetric Difference

• Commutative: A Δ B = B Δ A
• Associative: (A Δ B) Δ C = A Δ (B Δ C)
• Identity: A Δ ∅ = A
• Self-inverse: A Δ A = ∅