Symmetric Difference of Sets

To create new sets from old ones, pure mathematics uses a spread of methods. There are several methods for choosing specific elements from a collection while omitting others. The tip product is typically a collection that’s distinct from the originals. it is vital to own well-defined methods for constructing these new sets, like the union, intersection, and difference of two sets. The symmetric difference could be a set operation that’s possibly less well-known.

There are two sets shown within the upper image. The non – shaded region of the part is defined because the symmetric difference between the 2 sets and it’s denoted by A Δ B

Representation of Symmetric Difference :

Let us assume that set A and B are denoted by the symbol when denoting the symmetric difference between two sets

It is expressed mathematically as: A ∆ B = (A U B) – (A ∩ B)

Or, it may also be represented as A ∆ B = (A – B) U (B – A)

For the symmetric differences, this is often an identical expression. This representation is defined because the union of all the components present in a very but not in B with all the weather present in B but not in an exceedingly.One thing that both representations have in common, as we will see from the 2 different representations, is calculating the difference between the 2 sets. Not only is there a difference, but the equations are symmetric. By symmetric, we imply that we will modify the positions of the set A and B without affecting the end result.

The Difference between Sets vs Symmetric Difference

As the name implies, symmetric difference also performs a difference operation. So, how is it different from simply a group difference? Take a glance at the word ‘Symmetric Difference’ to work out what I mean. The word symmetric’ implies that the procedure must be symmetric. The difference between sets, however, isn’t symmetric, as we know. The difference between sets A and B won’t be the identical because the difference between sets B and A.Symmetric differences are distinguished from set differences by their symmetric nature.

Commutative Property : within the mathematical realm, any operation is taken into account commutative if the order of the operands doesn’t impact the result. It’s one among the foremost fundamental features of variety of binary operations.The symmetric difference commutative property asserts that: ‘The symmetric difference between A and B is adequate the symmetric difference between A and B.’So, if we’ve two sets, A and B, we are able to write this mathematically:

A ∆ B = B ∆ A

Associativity Property: Associativity means you’ll be able to move or reposition the parenthesis without affecting the equation’s outcomes.The associative law of symmetric difference, on the opposite hand, asserts that moving parenthesis about in any expression of sets employing symmetric difference operation has no effect on the outcomes.The result won’t be littered with grouping sets for symmetric differences.So, for example we’ve got three sets: A, B, and C. The mathematical expression for the associative property of symmetric difference is:

(A ∆ B) ∆ C = A ∆ (B ∆ C)

Few Examples:

Example 1:

Let A and B be two sets

A={2,4,6,8}

B={2,3,4,5}

We will see the step by step solution of the matter.

(A U B)={2,3,4,5,6,8}

The second step is to calculate the intersection of two sets. The intersection of set A and B will be:

(A ∩ B) = {2,4}

We take the difference between these two sets in the next step: A U B and A ∩ B.

(A U B) – (A ∩ B) = {3,5,6,8}

Example 2:

A = {10, 2, 4, 7}

B = {5, 6, 7, 10}

A ∆ B = (A U B) – (A ∩ B)

(A U B) = {2,4,5,6,7,10}

(A ∩ B) = {7,10}

(A U B) – (A ∩ B) = {2,4,5,6} = A ∆ B