por The closure property of algebra is a phenomenon that relates two elements of a set with an operation, where the necessary condition is that, after the 2 elements have been processed under said operation, the result also belongs to the initial set.
For example, if we take even numbers as a set and addition as an operation, we obtain a closure of that set with respect to addition. This is because the sum of 2 even numbers will always result in another even number, thus fulfilling the lock condition.
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Characteristics
There are many properties that determine spaces or algebraic fields, such as structures or rings. However, the closure property is one of the best known in basic algebra.
Not all applications of these properties are based on phenomena or numerical elements. Many everyday examples can be worked from a pure algebraic-theoretical approach.
An example can be the citizens of a country who assume a legal relationship of any kind, such as a business partnership or marriage, among others. After this operation or management is carried out, they continue to be citizens of the country. Thus citizenship and management operations with respect to two citizens represent a lock.
numerical algebra
Regarding numbers, there are many aspects that have been the subject of study in different currents of mathematics and algebra. From these studies, a large number of axioms and theorems have emerged that serve as a theoretical basis for contemporary research and work.
If we work with the numerical sets we can establish another valid definition for the property of lock. A set A is said to be the closure of another set B if A is the smallest set that contains all the sets and operations contained in B.
Demonstration
The closure proof is applied for elements and operations present in the set of real numbers R.
Let A and B be two numbers that belong to the set R, the closure of these elements is defined for each operation contained in R.
Addition
– Addition: ∀ TO ˄ B. ∈ R. → A+B=C ∈ R.
This is the algebraic way of saying that For all A and B that belong to the real numbers, the sum of A plus B is equal to C, which also belongs to the real numbers.
It is easy to check whether this proposition is true; it is enough to carry out the sum between any real number and verify if the result also belongs to the real numbers.
3 + 2 = 5 ∈ R.
-2 + (-7) = -9 ∈ R.
-3 +1/3 = -8/3 ∈ R.
5/2 + (-2/3) = 11/6 ∈ R.
It is observed that the closure condition is fulfilled for the real numbers and the sum. In this way it can be concluded: The sum of real numbers is an algebraic closure.
Multiplication
– Multiplication: ∀ TO ˄ B. ∈ R. → TO . B=C ∈ R.
For all A and B that belong to the reals, it is found that the multiplication of A by B is equal to C, which also belongs to the reals.
When verifying with the same elements of the previous example, the following results are observed.
3 x 2 = 6 ∈ R.
-2 x (-7) = 14 ∈ R.
-3 x 1/3 = -1 ∈ R.
5/2 x (-2/3) = -5/3 ∈ R.
This is enough evidence to conclude that: The multiplication of real numbers is an algebraic closure.
This definition can be extended to all operations on real numbers, although we will find certain exceptions.
Special cases in R
Division
As the first special case, division is observed, where the following exception is observed:
∀ TO ˄ B. ∈ R. → A/B ∉ R. ↔ B = 0
For all A and B that belong to R, it follows that A between B does not belong to the reals if and only if B is equal to zero.
This case refers to the restriction of not being able to divide by zero. Since zero belongs to the real numbers, then it follows that: heThe division is not a lock in the real ones.
filing
There are also potentiation operations, more specifically those of rooting, where there are exceptions for radical powers of even index:
; with n pair
For all A that belongs to the reals, the nth root of A belongs to the reals, if and only if A belongs to the positive reals joined to a set whose only element is zero.
In this way it is denoted that the even roots only apply to positive reals and it is concluded that the exponentiation is not a closure in R.
Logarithm
In a homologous way it is appreciated for the logarithmic function, which is not defined for values less than or equal to zero. To check if the logarithm is a closure of R, proceed as follows:
For every A that belongs to the reals, the logarithm of A belongs to the reals, if and only if A belongs to the positive reals.
By excluding the negative values and the zero that also belong to R, it can be affirmed that:
The logarithm is not a lock on the real numbers.
examples
Check the lock for addition and subtraction of natural numbers:
Sum in N
The first thing is to check the lock condition for different elements of the given set, where if it is observed that any element breaks the condition, the existence of a lock can be automatically denied.
This property holds for all possible values of A and B, as observed in the following operations:
1 + 3 = 4 ∈ No.
5 + 7 = 12 ∈ No.
1000 + 10000 = 11000 ∈ No.
There are no natural values that break the closure condition, so it is concluded:
The sum is a closure on N.
Subtraction in N
Natural elements capable of breaking with the condition are sought; A – B belongs to the natural ones.
Operating it is easy to find pairs of natural elements that do not meet the closure condition. For example:
7 – 10 = -3 ∉ to N
Thus we can conclude that:
Subtraction is not a closure of the set of natural numbers.
proposed exercises
1-Prove if the closure property is fulfilled for the set of rational numbers Q, for the operations addition, subtraction, multiplication and division.
2-Explain if the set of real numbers is a closure of the set of integers.
3-Determine which numerical set can be a lock of the real numbers.
4-Demonstrate the closure property for the set of imaginary numbers, with respect to addition, subtraction, multiplication and division.
References
Pure Mathematics Overview: the bourbakist choice. Jean Dieudonné. reverse1987.
Algebraic number theory. Alejandro J. Diaz Barriga, Ana Irene Ramirez, Francis Thomas. National Autonomous University of Mexico1975.
Linear Algebra and its Applications. Sandra Ibeth Ochoa Garcia, Eduardo Gutierrez Gonzalez.
Algebraic structures V: body theory. Hector A. Merklen. Organization of American States, General Secretariat1979.
Introduction to commutative algebra. michael francis atiyah, IG MacDonald. reverse1973.