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I know that the $\oplus$ prize is the exclusive or symbol however I don"t understand exactly how two the the very same sets space XOR to every other.
Sorry if this is a very basic question.
The prize $\oplus$ means direct sum.
The straight sum of 2 abelian teams $G$ and $H$ is the abelian group on the collection $G\times H$ (cartesian product) with the team operation given by $(g,h) + (g",h") = (g+g",h+h")$.
You might well have seen this team denoted $G\times H$ and also indeed, as lengthy as the number of terms is finite, the straight sum and direct product that abelian teams are isomorphic.
More precisely, the straight sum is the coproduct in the category of abelian groups, while the straight product is the product.
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