Question

Take into account the setstudent submitted image, transcription available belowstudent submitted image, transcription available belowcreated by the subsequent regulations:

student submitted image, transcription available below(1,0,0), (0,1,0), (0,0,3)student submitted image, transcription available below

student submitted image, transcription available belowIf (x,y,z), (a,b,c)student submitted image, transcription available below, then (xa,yb,zc)student submitted image, transcription available below

student submitted image, transcription available belowIf (x,y,z)student submitted image, transcription available belowthen (y,x,z)student submitted image, transcription available below

Proposition. V(x,y,z)student submitted image, transcription available below, 3|(xyz)

Why might using structural induction to try to verify this proposition be a good idea?

(a) S is a set, therefore structural induction makes sense.

Answer 1

Structural induction is an appropriate technique for proving a probabilistic statement for a recursively defined set, by confirming it for base cases and showing it holds for operations that generate new set elements.

Step-by-step explanation:

The question posed suggests a proposition that for any triplet of integers (x, y, z) formed by given rules, the product xyz is divisible by 3. Structural induction is a suitable method to use here because it systematically verifies the truth of a proposition for all elements of a recursively defined set. This aligns with how the set in question is created through the application of specific rules that define its structure. One starts by proving the proposition for the base cases, then assumes it holds for a general case, and then proves that it must hold after applying the generating rules. In the context of the given set, those rules involve element-wise multiplication and swapping of elements, which maintain the divisible by 3 property.