WHAT IS THE CLOSING PROPERTY? (WITH EXAMPLES) - MATHS - 2024

The closure property is a basic mathematical property that is fulfilled when a mathematical operation is performed with two numbers that belong to a specific set and the result of said operation is another number that belongs to the same set.

If we add the number -3 that belongs to the real numbers, with the number 8 that also belongs to the real ones, we get as a result the number 5 that also belongs to the real ones. In this case we say that the closure property is satisfied.

Generally this property is defined specifically for the set of real numbers (ℝ). However, it can also be defined in other sets such as the set of complex numbers or the set of vector spaces, among others.

In the set of real numbers, the basic mathematical operations that satisfy this property are addition, subtraction and multiplication.

In the case of division, the closure property only fulfills the condition of having a denominator with a value other than zero.

Closing property of addition

The addition is an operation by means of which two numbers are united in one. The numbers to be added are called Addends while their result is called Sum.

The definition of the closure property for addition is:

• Being a and b numbers belonging to ℝ, the result of a + b is a unique one in ℝ.

Examples:

(5) + (3) = 8

(-7) + (2) = -5

Closing property of subtraction

Subtraction is an operation in which we have a number called a Minuend, from which a quantity represented by a number known as a Subtrand is extracted.

The result of this operation is known by the name of Subtraction or Difference.

The definition of the closure property for subtraction is:

• Being a and b numbers belonging to ℝ, the result of ab is a single element in ℝ.

Examples:

(0) - (3) = -3

(72) - (18) = 54

Closing property of multiplication

Multiplication is an operation in which from two quantities, one called Multiplying and the other called Multiplier, a third quantity called Product is found.

In essence, this operation involves the consecutive addition of the Multiplying as many times as the Multiplier indicates.

The closure property for multiplication is defined by:

• Being a and b numbers belonging to ℝ, the result of a * b is a single element in ℝ.

Examples:

(12) * (5) = 60

(4) * (-3) = -12

Clausurative property of division

Division is an operation in which, from a number known as Dividend and another called Divisor, another number known as Quotient is found.

In essence, this operation implies the distribution of the Dividend in as many equal parts as indicated by the Divisor.

The closing property for division only applies when the denominator is nonzero. According to this, the property is defined like this:

• Being a and b numbers belonging to ℝ, the result of a / b is a single element in ℝ, if b ≠ 0

Examples:

(40) / (10) = 4

(-12) / (2) = -6

References

1. Baldor A. (2005). Algebra. Editorial group patria. Mexico. 4ed.
2. Camargo L. (2005). Alpha 8 with standards. Editorial Norma SA Colombia. 3ed.
3. Frias B. Arteaga O. Salazar L. (2003). Fundamental Mathematics For Engineers. National university of Colombia. Manizales, Colombia. 1ed.
4. Fuentes A. (2015). Algebra: a Mathematical Analysis Preliminary to Calculus. Colombia.
5. Jimenez J. (1973). Linear Algebra II with Applications in Statistics. National university of Colombia. Bogota Colombia.