Closure Definition Math

In mathematics, Closure refers to the likelihood of an operation on elements of a set. If something is closed, then it implies that if we conduct an operation on any two elements in a set, then the outcome of the operation is also in the set. If there exist elements in a set for which the outcome of an operation is not in the set, then the operation is not said to be closed.


Examples of Closure Property

Examples, take into account addition on natural numbers where natural numbers are the set and addition is the operation. Since a natural number added to another natural number will always be a natural number, and then we say that closed under addition or addition is closed on the natural numbers. Now considering the operation of division on natural numbers, we can see that for the two numbers 2 and 3, that outcome of 2 divided by 3 is 2/3. However, since 2/3 is not a natural number, then we notice that division on natural numbers is not closed.


Closure For Multiplication

The elements of a set of real numbers are closed under multiplication. If you perform multiplication of two real numbers, you will obtain another real number. There is no possibility of ever obtaining anything apart from another real number.

4*5 = 20

3½ * 2½ = 8 ¾

1.5 * 2.1 = 3.15


Closure For Addition

The elements of a set of real numbers are closed under addition. If you will add two real numbers, you will obtain another real number. There is no possibility of ever obtaining anything apart from another real number. 

5 + 12 = 17

3 ½ + 6 = 9 ½


Closed Under Addition and Closed Under Multiplication

If for every a ∈ S and b ∈ S, a + b ∈ S ; we say that S is closed under addition. The meaning of multiplication is analogous. Here are some examples:

The sets N, Z, Q and R are all closed under addition as well as multiplication.

The set of elements (0, 1) = {x: x ∈ R, 0 < x < 1} is closed under multiplication, but not addition. (0.6 + 0.7 = 1.3 > 1)

The set of the half integers Z/2 = {x: ∃ y ∈ Z(x = y/2)} is closed under addition, but not under multiplication (0.5 ⋅ 0.5 = 0.25 ∉ Z/2).


What Does Closed Under Addition Mean?

It is speaking of a set, (the mathematical theory) and the algebraic operation, here, addition. For any given two members of the set, addition will result in a value that is also a member of the set, provided that the set is closed.

Addition of the set of positive integers is closed due to the reason that the sum of any two integers is also an integer.

Addition of 2 digit integer numbers is not closed, under addition for various reasons. Take for example, 60 + 61 = 121, a 3 digit number. Why is that an issue? Because our original set was only the two-digit numbers. In addition, members of the beginning set can be combined to produce outcomes outside of the set.


Which Set of Numbers is Closed Under Subtraction?

Considering subtraction, Integers give closure under subtraction, whereas whole numbers do not. At a certain point, people were confronted with the issue of having to divide one thing among more than one person. From this conflict came into being the set of rational numbers. Positive numbers do not provide closure under subtraction because picking a bigger number to subtract from a smaller one results in a negative number result.


What are Closed Numbers?

In mathematics, the natural numbers turn to be "closed" under addition and multiplication. A set is closed (under an operation) if and only if the operation on any two elements of the set yields another element of a similar set. If the operation yields even one element outside of the set, the operation does not provide the closure.

Simply a set is said to be closed under an operation if conducting that operation on members of the set always yields a member of that set. For example, the positive integers are not closed under subtraction, but are under addition: 1 − 2 is not a positive integer despite both 1 and 2 are positive integers.

FAQs (Frequently Asked Questions)

Q1. Is a Set of Irrational Numbers Closed Under Algebraic Operations?

Answer: No - for a set of numbers to be closed under an assigned operation then any pair of numbers with that operation shall outcome in a number within that set; or in reverse. If you could find a pair of numbers in the set where the operation outcomes in a number not in the set then that set is not closed over that operation.

Q2. How is the Set of Irrational Numbers not Closed Under Addition, Subtraction and Multiplication?

Answer: Below is a description of how and why the set of irrational numbers is not closed under addition, subtraction and multiplication.

  • Addition

π + (−π) = 0

π and −π are both irrational but 0 is not irrational. Therefore, irrationals are not closed over addition.

  • Subtraction

 Taking into account different values - just to prove the point

-√2-(-√2) = 0

-√2 is irrational but 0 is not irrational. Therefore, irrationals numbers are not closed over addition.

  • Multiplication

-√3 ∗-√3 = 3

-√3  is irrational but 3 is not irrational. ∴ Iirrationals are not closed over multiplication.

  • Division

√5/√5 = 1  

√5 is irrational but 1 is not irrational. Therefore, irrationals are not closed over division.

As you can observe that we have found pairs of values for each operation that produces a non-irrational outcome, so thus the set of irrationals cannot be closed under any of the operations of addition, subtraction and multiplication and division.