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 April 8th, 2017, 01:59 AM #1 Newbie   Joined: Jul 2012 Posts: 5 Thanks: 0 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!
 April 8th, 2017, 12:16 PM #2 Senior Member     Joined: Sep 2015 From: USA Posts: 2,009 Thanks: 1041 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.
 April 9th, 2017, 11:02 PM #3 Newbie   Joined: Jul 2012 Posts: 5 Thanks: 0 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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