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I recognize that the $\oplus$ price is the exclusive or symbol yet I don"t understand how two of the same sets are XOR to each other.

Sorry if this is a very simple question.

The symbol $\oplus$ method direct sum.

The direct sum of two abelian teams $G$ and $H$ is the abelian team on the collection $G\times H$ (cartesian product) v the team operation provided by $(g,h) + (g",h") = (g+g",h+h")$.

You might well have actually seen this group denoted $G\times H$ and also indeed, as long as the variety of terms is finite, the direct sum and direct product that abelian groups are isomorphic.

More precisely, the direct sum is the coproduct in the classification of abelian groups, while the direct product is the product.

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