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April 8th, 2017, 01:59 AM   #1
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How summing groups of elements?

Summing groups of elements is probably one of the mathematical things that people apply the most.
But I wonder how this is done mathematically in a correct way.
Examples:
  • {1 apple} “adding to” {1 apple} = {2 apples}
  • {1 apple, 2 pears} “adding to” {2 apples, 1 banana} = {3 apples, 2 pears, 1 banana}
  • {1 apple} “removing from” {1 apple} = {}

Notes:
  • This are all valid sets and the concept of multisets is completely irrelevant for this question. See definitions of sets and multisets.
  • “+” and “-“ are not defined for sets => no solution
  • “U” a union is also not useful. {1 apple} U {1 apple} = {1 apple} (and not 2 apples)

So which mathematical technique is used to sum groups in a in a correct way?
Thanks in advance!
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April 8th, 2017, 12:16 PM   #2
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I would just use vector addition.

Each element of the vector corresponds to the number of a unique type of fruit or whatever. You can make the elements mean whatever you like as long as they are unique descriptors.

letting $v=(apples, pears, bananas)$ we have

$(1,2,0)+(2,0,1) = (3,2,1)$

$(1,0,0)+(1,0,0) = (2,0,0)$

etc.
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April 9th, 2017, 11:02 PM   #3
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Thanks!
Vector addition is indeed a way (maybe the only) to do this. Although I have some doubts about the axiom about the existence of an inverse element (required for a vector space). I can imagine that some mathematicians will say that they have never seen a negative apple. Unless of course that removing an apple can be considered as inverse element.
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