why $\alpha ,\beta,\gamma$ :$(\beta + \gamma)\alpha = \beta\alpha + \beta \gamma$ doesn't hold?

Question

this rule doesn't hold for all ordinals $\alpha ,\beta,\gamma$ :$(\beta + \gamma)\alpha = \beta\alpha + \beta \gamma$. I tested many examples but all of them holds for it ! does this hold ? $(B \cup C )\times A = (B \times A ) \cup(C\times A)$ for the their sets? why the above rule doesn't hold ? please prove for me ! It hold for many finite and infinite sets I used !

Answer 1

Here's a counter example: $$(1+1) \cdot \omega = 2 \cdot \omega = \omega \neq \omega + \omega = \omega \cdot 2$$

• To see that $2 \cdot \omega = \omega$, recall that $2 \cdot \omega = \lim_{n < \omega} 2 \cdot n$. This is clearly equal to $\omega$.
• $\omega + \omega \neq \omega$ because there's an element $\alpha \in \omega + \omega$ that has an infinite number of elements less than $\alpha$. This doesn't happen in $\omega$.