The symmetric difference of two sets can be defined as the set of elements that belong to one of the initial sets, but do not belong to both. For example, the symmetric difference of {4,2,3,8} and {1,2,3,7} is {1,4,8}.
What is set theory
The development of set theory is due to the German mathematician Georg Cantor, in the mid-nineteenth century. Currently, set theory is part of mathematical logic and focuses on the study of the properties and relationships between sets.
A set is an abstract collection of elements, which are considered objects themselves. Sets and the operations performed with them constitute one of the most basic tools in the formulation of mathematical theories.
Set theory also includes all other mathematical objects and structures, such as numbers, geometric figures, functions, and others. Even set theory is also an object of study by itself and is almost always accompanied by the application of logic.
What are the sets
Sets are defined as a series of well-defined objects, often called elements, that differ from each other. Sometimes a set is expressed according to the property or properties that its elements satisfy.
A set can have a finite or infinite number of elements. In mathematics it is common to name elements with lowercase letters and sets with uppercase letters. Also, sets and their operations can be represented graphically with Venn diagrams.
Venn diagram illustrating the intersection between two sets
Set Operations
Basic operations can be performed on sets and their elements that are similar to arithmetic operations, and make up what is known as the algebra of sets. These operations are:
- Intersection : this operation results in a set A ∩ B where are all the elements that sets A and B have in common.
- Symmetric Difference: The symmetric difference of two sets A and B is the set A Δ B . This includes all elements that are part of A or B, but not both at the same time.
- Difference : the operation between two sets is known as difference, in this case, A and B. The set A B is obtained, which includes all the elements of set A that do not belong to set B.
- Complement : The complement of a set A is the set A∁ . This contains all the elements that do not belong to A.
- Cartesian Product : This operation results in the set A × B where its elements are ordered pairs. The first element belongs to A and the second belongs to B.
- Union: is the union of two or more sets. For example, the union of the sets A and B is expressed as the set A ∪ B . This contains all the elements that were in them.
The symmetrical difference: concept and characteristics
As mentioned above, the symmetric difference of two sets is another set where the elements that belong to the initial sets are included, but are not in both at the same time. For example, the symmetric difference of the sets A={2,5,3} and B={4,2,3,7} is the set A Δ B={4,5,7}.
That is, the symmetric difference is an operation where the difference between two or more sets is observed.
Other examples to understand the symmetric difference are:
- If we have the sets A={1,2,3,4,5} and B={2,4,6}, then the symmetric difference between these sets is the set A ∆ B={1,3,5,6 }.
- The symmetric difference of the sets R={a, b, c, d } and S={a, b, e, f } is the set R ∆ S={c, d, e, f}.
The symmetric difference between A and B can also be defined as the difference between the union and the intersection of A and B. This is expressed: A ∆ B = (A ∪ B ) – (A ∩ B).
Another equivalent way to express the symmetric difference using the union and intersection operations is: (A – B) ∪ (B – A). Here it can be seen that the symmetric difference is the set of elements in A but not in B, or in B but not in A. Thus, those elements at the intersection of A and B are excluded.
Symmetric Difference Properties
From the concepts mentioned about the symmetric difference, different properties can be deduced:
- The symmetric difference of a set with respect to itself is the empty set: A Δ B = Ø
- Therefore, the symmetric difference of a set A with the empty set is the same set A: A Δ Ø = A
- The symmetric difference of a set and one of its subsets is the difference between them: B ⊆ A → A Δ B= A B
- And the symmetric difference of the sets A Δ B and C is the same as that of the sets A Δ B and C. This is expressed: (A Δ B) Δ C = A Δ (B Δ C)
- Likewise, the symmetric difference of the sets A and B is equal to the symmetric difference of the sets B and A. Which is represented as follows: A Δ B = B Δ A
Bibliography
- Morra, J. Topic 11. Basic concepts of set theory. Algebraic structures . (2020, Kindle edition. Spain. B085WBRJNC.
- López Mateos, M. Sets, Logic and Functions. (2019, 2nd edition). Spain. Manuel Lopez Mateos.
- Uzcátegui Aylwin, C. An introduction to the descriptive theory of sets. (2020). Spain. Uniande Editions.
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