Can Cardinal Arithmetic Mimic Distributive Property?

[evinda]

Hi! (Nerd)

I want to prove for any cardinal numbers $m,n,p$ it holds that:

$$m \cdot (n+p)=m \cdot n+m \cdot p$$

Could we prove this using induction on [m] m [/m] ?

Or could we maybe show that $A \times (B \cup C)=(A \times B) \cup (A \times C)$ where $card(A)=m, card(B)=n, card(C)=p$ ? (Thinking)

 

[Valkarie]

Yes, both of these methods are valid ways to prove this statement. To prove it with induction, you would start by showing that the statement is true for $m=1$, then assume that it is true for $m=k$ and show that it is true for $m=k+1$. To prove it using the set equation, you would expand out each side of the equation and show that they are equivalent.