@R.vanDobbendeBruyn Thanks! -- I edited out the redundant factor of 2 from the example.

@R.vanDobbendeBruyn To prove that a given triple does not to satisfy the condition, one can use the algorithm described in: B. M. M. de Weger, Solving exponential Diophantine equations using lattice basis reduction algorithms, J. Number Theory 26 (1987), no. 3, 325–367. An example is $\{\{5\},\{13\},\{17\}\}$.

@R.vanDobbendeBruyn Well -- it is clear that not all numbers can be odd, as the sum of two odd numbers is even. On the other hand, not all numbers need to be even. E.g. we have $\{\emptyset, \{3\}, \{5\}\} \in \Delta$ since $2 + 3 = 5$.

@FedericoPoloni If you like, you can post a CW answer along the lines of Jochen Glueck's comment.

@JochenGlueck This question has presently no close votes, 3 upvotes and no downvotes. Therefore, I don't see a reason not to post a proper answer.

I still learned these letters in elementary school -- but as old / historic, and not as usual font.

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Please be aware that every edit of a question or of one of its answers bumps the thread to the front page. This has happened for this thread already more than 20 times, and this is a nuisance for other users. Please refrain from unnecessary edits to your posts. -- Usually, the vast majority of minor edits can be avoided by writing and proofreading a question or an answer (or any update to such) carefully before posting it.