8 junio, 2024

Closure property: what it is and examples

What is the closing property?

The cloisterable property is a basic mathematical property that holds when a mathematical operation is performed on two numbers that belong to the same specific set, and the result of said operation is another number that belongs to the same set.

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

Generally, this property is defined specifically for the set of real numbers (ℝ). However, it can also be defined on 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 is only fulfilled on the condition of having a denominator with a value other than zero. What happens is that in the division, many times, the quotient of whole numbers is not a whole number: 25 / 3 = 8.33333.

It is said to be closing because the operations (addition, subtraction, multiplication or division, with their conditions) are closed on the set of real numbers.

Closing Property of Addition

Addition is an operation by which two numbers are joined into one. The numbers to be added are called addends, while their result is called the sum.

The definition of the closure property for addition is:

Being a and b numbers that belong to ℝ, the result of a+b is unique in ℝ.

Examples:

(5) + (3) = 8

(-7) + (2) = -5

(-10) + (-4) = 14

Closing property of subtraction

Subtraction is an operation in which there is a number called a minuend, from which an amount represented by a number known as a subtrahend 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 that belong to ℝ, the result of ab is a unique 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 the multiplicand and the other called the multiplier, a third quantity called the product is found.

In essence, this operation implies the consecutive addition of the multiplicand as many times as the multiplier indicates.

The closure property for multiplication is defined by:

Being a and b numbers that belong to ℝ, the result of a*b is a unique element in ℝ.

Examples:

(12) * (5) = 60

(4) * (-3) = -12

Closing 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 closure property for division only applies when the denominator is different from zero. According to this, the property is defined like this:

Being a and b numbers that belong to ℝ, the result of a/b is a unique element in ℝ, if b≠0.

Examples:

(40) / (10) = 4

(-12) / (2) = -6

(25) / (5) = 5

In other cases: (18) / (5) = 3.6 (it does not fulfill the closing property because the quotient is a decimal number).

Examples of the cloister property

149 + 43 + 67 = 326 (addition) -98 + 78 = -20 (addition) 125 – 75 = 50 (subtraction) 12*4 = 48 (multiplication) 100 / 50 = 2 (division)

References

Algebra. Home publishing group. Mexico.
Alpha 8 with standards. Editorial Norma SA Colombia.

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