Sets where you can't do basic arithmetic operations

[meer]

Hi,
For a set of real numbers, or integers and few more, we can perform basic arithmetic operations like x+y, x-y, xy, x/y.
what's example of a set/s where we can't perform these basic operations?

Thank you.

 

[Deleted member 4993]

Hi,
For a set of real numbers, or integers and few more, we can perform basic arithmetic operations like x+y, x-y, xy, x/y.
what's example of a set/s where we can't perform these basic operations?

Thank you.

For which number (y) - the operation of division (x/y) is NOT defined?

 

[Dr.Peterson]

Hi,
For a set of real numbers, or integers and few more, we can perform basic arithmetic operations like x+y, x-y, xy, x/y.
what's example of a set/s where we can't perform these basic operations?

Thank you.

Are you looking for a subset of the real numbers that is not closed under each operation? Or a set separate from the real numbers on which the operations are not defined at all? Or what? Please clarify your meaning of "can't perform".

What is the context of your question? If it is for a class, what topic is being studied? If not, what inspired your question?

 

[JeffM]

Hi,
For a set of real numbers, or integers and few more, we can perform basic arithmetic operations like x+y, x-y, xy, x/y.
what's example of a set/s where we can't perform these basic operations?

Thank you.

The set of types of humanly edible fruits.

[math]\text{plum} \times \text {raspberry} \text { is meaningless nonsense.}[/math]
By any chance are you asking whether some sets of numbers are not closed under one or more basic operations?

 

[topsquark]

Hi,
For a set of real numbers, or integers and few more, we can perform basic arithmetic operations like x+y, x-y, xy, x/y.
what's example of a set/s where we can't perform these basic operations?

Thank you.

When we consider sets with operations on them we usually require certain properties. Given a set G with an operation * on it
1) For all a and b in G then c = a*b is also in G. (Closure)
2) Given a, b, and c in G then a*(b*c) = (a*b)*c (Associativity)
3) There is a unique element e of G such that a*e = e*a = a (for every a in G) (Identity)
4) For each a in G there is a unique element [imath]a^{-1}[/imath] such that [imath]a*a^{-1} = a^{-1}*a = e[/imath] (Inverses)
(When * is actually more like an addition we use the symbol + for the group operation. Some groups have both a + and * operation defined on them. With a few more conditions we call this structure a field.)

If these conditions are met then the set G and operation * combine to form an object called a "group." Most objects of Mathematical interest are groups. We can add "structure" to this as well... Commutative groups have the additional property a*b = b*a.

So, to address your question, it doesn't really make a lot of sense to define an operation such that your set would not be closed. If we can define a*b then we usually just add the element c = a*b to the set.

Now, there are some sets where division is not impossible, but difficult to work with. These are called "modular groups." (These are sometimes called "remainder groups.") For example, consider an old fashioned clock with hands and everything. It reads time from Midnight = 0 to Noon = 12, and then resets to 0. So if we add the two times, 3 + 4 we get 7, and if we add the two times, 7 + 8 we get 7 + 8 = 15 which resets to 3 (because 7 AM plus 8 hours is 3 PM.) We can multiply here, too. 3* 3 = 9 and 5*4 = 20 = 8. We can define division but because fractions are forbidden (they don't belong to the set) we run into some problems with some numbers. The multiplicative inverse of 5 is 5 because 5*5 = 25 = 1 (recall that an inverse is an element [imath]a^{-1}[/imath] such that [imath]a*a^{-1} = e[/imath].) So, for example, 7/5 would be defined to be [imath]7/5 =7*5^{-1} = 7*5 = 35 = 11[/imath]. (You can check this easily... if 7/5 = 11, then 11*5 = 55, but 55 resets to 7 so 11*5 = 7 as expected.) But how can we divide by, say, 2? There is no element in our set such that 2*a = 1. Yes, we can divide 10/2 and get 5. But we cannot legally do the process this way, we have to be able to do it for more than even numbers. How would we do 7/2, for example? 2, 3, 4, 6, 8, and 9 have no multiplicative inverses in this system so division is pretty dodgy. "Clock Math" is the group [imath]\mathbb{Z}_{12}[/imath]. Other modular groups work the same way and can be very useful.

-Dan

 

[meer]

Are you looking for a subset of the real numbers that is not closed under each operation? Or a set separate from the real numbers on which the operations are not defined at all? Or what? Please clarify your meaning of "can't perform".

What is the context of your question? If it is for a class, what topic is being studied? If not, what inspired your question?

Doc, Thank you for your reply
In precalculus or calculus, we usually start with functions and in the review of basics they give us examples of sets like real numbers, integers, natural numbers etc. where we can perform the basic operations of addition, subtraction multiplication and division. I requested for examples of sets (separate and independent of the stated sets) where we can't do these basic arithmetic operations. As an example JeffM replied (in the same thread above) set of edible fruits, like wise it could be set of colors. So I am looking for such an example/s that belongs to the modern technological world and is practicality applicable in some high level academic area.

Regards

 

[meer]

When we consider sets with operations on them we usually require certain properties. Given a set G with an operation * on it
1) For all a and b in G then c = a*b is also in G. (Closure)
2) Given a, b, and c in G then a*(b*c) = (a*b)*c (Associativity)
3) There is a unique element e of G such that a*e = e*a = a (for every a in G) (Identity)
4) For each a in G there is a unique element [imath]a^{-1}[/imath] such that [imath]a*a^{-1} = a^{-1}*a = e[/imath] (Inverses)
(When * is actually more like an addition we use the symbol + for the group operation. Some groups have both a + and * operation defined on them. With a few more conditions we call this structure a field.)

If these conditions are met then the set G and operation * combine to form an object called a "group." Most objects of Mathematical interest are groups. We can add "structure" to this as well... Commutative groups have the additional property a*b = b*a.

So, to address your question, it doesn't really make a lot of sense to define an operation such that your set would not be closed. If we can define a*b then we usually just add the element c = a*b to the set.

Now, there are some sets where division is not impossible, but difficult to work with. These are called "modular groups." (These are sometimes called "remainder groups.") For example, consider an old fashioned clock with hands and everything. It reads time from Midnight = 0 to Noon = 12, and then resets to 0. So if we add the two times, 3 + 4 we get 7, and if we add the two times, 7 + 8 we get 7 + 8 = 15 which resets to 3 (because 7 AM plus 8 hours is 3 PM.) We can multiply here, too. 3* 3 = 9 and 5*4 = 20 = 8. We can define division but because fractions are forbidden (they don't belong to the set) we run into some problems with some numbers. The multiplicative inverse of 5 is 5 because 5*5 = 25 = 1 (recall that an inverse is an element [imath]a^{-1}[/imath] such that [imath]a*a^{-1} = e[/imath].) So, for example, 7/5 would be defined to be [imath]7/5 =7*5^{-1} = 7*5 = 35 = 11[/imath]. (You can check this easily... if 7/5 = 11, then 11*5 = 55, but 55 resets to 7 so 11*5 = 7 as expected.) But how can we divide by, say, 2? There is no element in our set such that 2*a = 1. Yes, we can divide 10/2 and get 5. But we cannot legally do the process this way, we have to be able to do it for more than even numbers. How would we do 7/2, for example? 2, 3, 4, 6, 8, and 9 have no multiplicative inverses in this system so division is pretty dodgy. "Clock Math" is the group [imath]\mathbb{Z}_{12}[/imath]. Other modular groups work the same way and can be very useful.

-Dan

Appreciate the detailed reply. Thank you.

 

[JeffM]

Doc, Thank you for your reply
In precalculus or calculus, we usually start with functions and in the review of basics they give us examples of sets like real numbers, integers, natural numbers etc. where we can perform the basic operations of addition, subtraction multiplication and division. I requested for examples of sets (separate and independent of the stated sets) where we can't do these basic arithmetic operations. As an example JeffM replied (in the same thread above) set of edible fruits, like wise it could be set of colors. So I am looking for such an example/s that belongs to the modern technological world and is practicality applicable in some high level academic area.

Regards

The ideas of sets, operations, and functions are not limited to numbers. In fact, I suspect that the idea of sets underlies almost all human thought. What, for example, are nouns, verbs, and adjectives except names for sets.

Norman Campbell wrote two essays in which he pointed out that the applicability of numbers and arithmetic operations to the physical world must in principle be verified by experiment. Part of that explanation involved showing that one application of numbers to the modern world, namely the use of numbers as an inexhaustible source of unique labels or names, does not permit the meaningful use of the arithmetic operations. We can go through the mechanical operation of multiplying social security numbers, but the result is not a valid social security number. The set of social security numbers is not closed under the arithmetic operations.

Topsquark pointed out that there is a branch of mathematics (abstract algebra) that has generalized some key features of arithmetic operations to operations on sets that are not numbers, for example rotations of shapes.

But it is always a matter to be determined whether operations that are generalizations of arithmetic operations can be applied to sets, even sets of numbers. Closure is a really important property. But the set of odd numbers is not closed under any of the arithmetic operations except multiplication.