What Does 'Closed Under Addition' Mean in Mathematics?

In summary, "closed under addition" means that the sum of two integers is an integer. This is not true for other operations such as division or square root extraction.
  • #1
garyljc
103
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What does it means by closed under addition
For eg : As S is closed under addition
S = {gn : n is a member of integers}
could anyone elaborate more on this and gimme some example ?

Does it mean that when something is closed under addition , we only consider addition and nothing else ?
 
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  • #2


garyljc said:
What does it means by closed under addition
For eg : As S is closed under addition
S = {gn : n is a member of integers}
could anyone elaborate more on this and gimme some example ?

Does it mean that when something is closed under addition , we only consider addition and nothing else ?

"Closed under addition" means that the sum of two integers is an integer.


Note, for example, that the integers are NOT closed under the operation of division, or for that matter, closed under square root extraction.
 

Related to What Does 'Closed Under Addition' Mean in Mathematics?

1. What does it mean for a set to be "closed under addition"?

Being closed under addition means that when two elements from the set are added together, the result is also an element of the set.

2. Why is it important for a set to be closed under addition?

It is important for a set to be closed under addition because it allows for all possible combinations of elements within the set to be valid and meaningful. This is essential in mathematical operations and proofs.

3. How can I determine if a set is closed under addition?

To determine if a set is closed under addition, you can perform addition operations on all possible combinations of elements within the set and check if the results are also elements of the set. If all combinations result in elements of the set, then the set is closed under addition.

4. What happens if a set is not closed under addition?

If a set is not closed under addition, it means that there are certain combinations of elements within the set that, when added together, do not result in an element of the set. This can lead to inconsistencies and errors in mathematical operations and proofs.

5. Can a set be closed under addition but not multiplication?

Yes, a set can be closed under addition but not multiplication. This means that while all possible combinations of elements within the set result in an element of the set when added together, they may not necessarily result in an element of the set when multiplied together.

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