Let α and β be cardinal numbers. The cardinal sum of α and β, denoted α + β, is the cardin...

Let α and β be cardinal numbers. The cardinal sum of α and β, denoted α + β, is the cardinal |A ∪ B|, where A and B are disjoint sets such that |A| = α and |B| = β.

(a) Prove that the sum is well-defined. that is, if |A| = |C|, |B| = |D|, A ∩ B = ∅, and C ∩ D = ∅, then |A ∪ B| = |C ∪ D|.

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(b) Prove that the sum is commutative and associative. That is, for any cardinals α, β, and γ, we have α + β = β + α and α + (β + γ) = (α + β) + γ

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(c)

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(d)

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(e)

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(f) show that c + c = c.