Decidable product with an undecidable projection I got some problems with building a set, which should looks like this: $S = A\times B \subset N \times N $, where $S$&nbsp;is decidable but $A$&nbsp;is undecidable. 

Could somebody give me a clue how to actually do this?
Let $A$ be set of Turing machines that halt, which is clearly undecidable. Now $B$ needs to be something that helps you decide $A$ (a certificate if you will, in the sense of a certificate for a $NP$ problem). Hint: if a TM halts, it halts in a finite number of steps.