Is the set of negative integers for subtraction closed?

All negative numbers can be added to one another and remain negative. Consider the integers $x$ and $y$, where $x,y<0$. Then take $x+y$,

then I want to say that having both x and y be negative, means that their sum is negative, but that seems too simple... like I'm using the fact I'm trying to prove to prove it.

"Closure" is a property which a set either has or lacks with respect to a given operation. A set is closed with respect to that operation if the operation can always be completed with elements in the set.

For example, the set of even natural numbers, 2, 4, 6, 8,..., is closed with respect to addition because the sum of any two of them is another even natural number. It is not closed with respect to division because the quotients 6/2 and 4/8, for instance, cannot be computed without using odd numbers or fractions.

Knowing the operations for which a given set is closed helps one understand the nature of the set. Thus one knows that the set of natural numbers is less versatile than the set of integers because the latter is closed with respect to subtraction, but the former is not. Similarly one knows that the set of polynomials is much like the set of integers because both sets are closed under addition, multiplication, negation, and subtraction, but are not closed under division.

Particularly interesting examples of closure are the positive and negative numbers. In mathematical structure these two sets are indistinguishable except for one property, closure with respect to multiplication. Once one decides that the product of two positive numbers is positive, the other rules for multiplying and dividing various combinations of positive and negative numbers follow. Then, for example, the product of two negative numbers must be positive, and so on.

The lack of closure is one reason for enlarging a set. For example, without augmenting the set of rational numbers with the irrationals, one cannot solve an equation such as x2 = 2, which can arise from the use of the pythagorean theorem. Without extending the set of real numbers to include imaginary numbers, one cannot solve an equation such as x 2 + 1= 0, contrary to the fundamental theorem of algebra.

Closure can be associated with operations on single numbers as well as operations between two numbers. When the Pythagoreans discovered that the square root of 2 was not rational, they had discovered that the rationals were not closed with respect to taking roots.

Although closure is usually thought of as a property of sets of ordinary numbers, the concept can be applied to other kinds of mathematical elements. It can be applied to sets of rigid motions in the plane, to vectors, to matrices, and to other things. For example, one can say that the set of three-by-three matrices is closed with respect to addition.

Closure, or the lack of it, can be of practical concern, too. Inexpensive, four-function calculators rarely allow one to use negative numbers as inputs. Nevertheless, if one subtracts a larger number from a smaller number, the calculator will complete the operation and display the negative number which results. On the other hand, if one divides 1 by 3, the calculator will display 0.333333, which is close, but not exact. If an operation takes a calculator beyond the numbers it can use, the answer it displays will be wrong, perhaps significantly so.

Mathematicians are often interested in whether or not certain sets have particular properties under a given operation. One reason that mathematicians were interested in this was so that they could determine when equations would have solutions. If a set under a given operation has certain general properties, then we can solve linear equations in that set, for example.

There are several important properties that a set may or may not satisfy under a particular operation.� A property is a certain rule that holds if it is true for all elements of a set under the given operation and a property does not hold if there is at least one pair of elements that do not follow the property under the given operation.  

Talking about properties in this abstract way doesn't really make any sense yet, so let’s look at some examples of properties so that you can better understand what they are. In this lecture, we will learn about the closure property.

The Property of Closure

A set has the closure property under a particular operation if the result of the operation is always an element in the set.� If a set has the closure property under a particular operation, then we say that the set is �closed under the operation.��

It is much easier to understand a property by looking at examples than it is by simply talking about it in an abstract way, so let's move on to looking at examples so that you can see exactly what we are talking about when we say that a set has the closure property:

First let�s look at a few infinite sets with operations that are already familiar to us:

a)      The set of integers is closed under the operation of addition because the sum of any two integers is always another integer and is therefore in the set of integers.�

b)      The set of integers is not closed under the operation of division because when you divide one integer by another, you don�t always get another integer as the answer.� For example, 4 and 9 are both integers, but 4 � 9 = 4/9.� 4/9 is not an integer, so it is not in the set of integers!

c)      The set of rational numbers is closed under the operation of multiplication, because the product of any two rational numbers will always be another rational number, and will therefore be in the set of rational numbers.� This is because multiplying two fractions will always give you another fraction as a result, since the product of two fractions a/b and c/d, will give you ac/bd as a result. The only possible way that ac/bd could not be a fraction is if bd is equal to 0. But if a/b and c/d are both fractions, this means that neither b nor d is 0, so bd cannot be 0.

d)      The set of natural numbers is not closed under the operation of subtraction because when you subtract one natural number from another, you don�t always get another natural number.� For example, 5 and 16 are both natural numbers, but 5 � 16 =� � 11.� � 11 is not a natural number, so it is not in the set of natural numbers!

Now let�s look at a few examples of finite sets with operations that may not be familiar to us:

e)      The set {1,2,3,4} is not closed under the operation of addition because 2 + 3 = 5, and 5 is not an element of the set {1,2,3,4}.�

We can see this also by looking at the operation table for the set {1,2,3,4} under the operation of addition:

+

1

2

3

4

1

2

3

4

5

2

3

4

5

6

3

4

5

6

7

4

5

6

7

8

The set{1,2,3,4} is not closed under the operation + because there is at least one result (all the results are shaded in orange) which is not an element of the set {1,2,3,4}.� The chart contains the results 5, 6, 7, and 8, none of which are elements of the set {1,2,3,4}!

f)        The set {a,b,c,d,e} has the following operation table for the operation *:

*

a

b

c

d

e

a

b

c

e

a

d

b

d

a

c

b

e

c

c

d

b

e

a

d

a

e

d

c

b

e

e

b

a

d

c

The set{a,b,c,d,e} is closed under the operation * because all of the results (which are shaded in orange) are elements in the set {a,b,c,d,e}.�

g)       The set {a,b,c,d,e} has the following operation table for the operation $: 

$

a

b

c

d

e

a

b

f

e

a

h

b

d

a

c

h

e

c

c

d

b

g

a

d

g

e

d

c

b

e

e

b

h

d

c

The set{a,b,c,d,e} is not closed under the operation $ because there is at least one result (all the results are shaded in orange) which is not an element of the set {a,b,c,d,e}.� For example, according to the chart, a$b=f.� But f is not an element of {a,b,c,d,e}!

Is the set of positive integers for subtraction closed?

What sets are closed under subtraction?

Why are negative numbers closed under subtraction?

How do you know if a set is closed under subtraction?