What Does 'Closed Under Addition' Mean in Mathematics?

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SUMMARY

"Closed under addition" refers to a set where the sum of any two elements within that set remains an element of the same set. In the discussion, the set S = {gn : n is a member of integers} exemplifies this property, as the sum of any two integers results in another integer. The integers are confirmed to be closed under addition, while they are not closed under operations such as division or square root extraction.

PREREQUISITES
  • Understanding of basic mathematical operations, specifically addition.
  • Familiarity with set theory concepts.
  • Knowledge of integer properties in mathematics.
  • Basic comprehension of closure properties in algebra.
NEXT STEPS
  • Research the closure properties of different mathematical sets, such as rational and real numbers.
  • Learn about closure under other operations, including multiplication and division.
  • Explore examples of sets that are not closed under specific operations.
  • Study the implications of closure properties in abstract algebra.
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Mathematics students, educators, and anyone interested in understanding the foundational concepts of set theory and closure properties in algebra.

garyljc
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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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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.
 

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