Let α and β be cardinal numbers. The cardinal product αβ is defined to be the cardinal |A...

Let α and β be cardinal numbers. The cardinal product αβ is defined to be the cardinal |A × B|, where |A| = α and |B| = β.

(a) prove that the product is well-defined. that is, if |A| = |C| and |B| = |D|, then |A × B| = |C × D|.

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(b) prove that the product is commutative and associative and that the distributive law holds. that is, for any cardinals α, β, and γ, we have αβ = βα, α(βγ) = (αβ)γ, and α(β + γ) = αβ + αγ.

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(c) show that 0α = 0 for any cardinal α.

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(d)

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(e)

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(f)