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Why does the operation of addition have to be defined as the joining of two non-intersecting...

Why does the operation of addition have to be defined as the joining of two non-intersecting sets?

Give an example of a problem where you combined two sets which are non-intersecting, and then give an example of adding two sets which have an intersection and explain why that is a problem.

(Think about two Venn Diagram with two sets each. One of the Venn diagrams has an intersection and one of them does not--how would the total number of elements in the two examples be calculated differently?)

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