I am solving the following equation out of a book
$⟦ A^{'}α{'}: α⊂A⟧$ and $⟦ B{'}β{'}: `β{'}⊂B⟧ $
(By the way, what does this notation mean? for any given alpha or beta follows that the given alpha or beta is part of the subset?)
$(∪{'}α{'} A{'}α{'}) × (∪{'}β{'} B{'}β{'}) = ∪(α,β) (A{'}α{'} × B{'}β{'})$
I understand why this is a correct equation. I am having however a hard time writing a correct logical proof. Here is my try.
$⊂$
let $(x,y) ∈ (∪α Aα) × (∪{'}β{'} B{'}β{'})$
$⇔ x∈ (∪α Aα) ^ y∈ (∪{'}β{'} B{'}β{'})$
$⇔ ∃α: x∈Aα ^ ∃β: y∈B{'}β{'}$
Let αo and βo be such indexes
$⇒ (x,y) ∈ ∪(α,β) (Aα × B{'}β{'})$ for $α{'}o{'}$ and $β{'}o{'}$
$⊃$
let $(x,y) ∈ ∪(α,β) (Aα × B{'}β{'})$
$⇔ ∃α∃{'}β{'}: (x,y) ∈ (Aα × B{'}β{'})$
$⇔ ∃α∃{'}β{'}: x∈ Aα ^ y∈B{'}β{'}$
(i was wondering if i could use this rule here? where as px and qx are attributes)
$∃x: p(x)^ q(x) ⇒ ∃x:p(x) ^ ∃x:q(x)$
Let αo and βo be such indexes
$⇒ (x,y) ∈ (∪α Aα) × (∪{'}β{'} B{'}β{'})$ for αo and βo
Is the proof correct ? Is there a better one?
Also how can I subscript and superscript when writing a question? Thank you!
