Also, if x is an element of Z mod 4, then x + 4 is not defined (since 4 is a number not an equivalence class mod 4. If you meant the equivalence class of 4 instead, then by the definition of modular arithmetic, y =x and the think you mean to say that if

]]>How did your students respond to the excercise>

]]>More deeply, though it may not be especially relevant for your present purpose: I think the fundamental insight here is that the product of an even integer & any other integer is even. (Proof: Write the former as 2m and the latter as n; then their product is 2mn, which is an integer multiple of 2, i.e., even.)

Corollary: Any even squared is even.

Corollary to corollary: Your original Theorem.

I expect that for these students the corollaries should actually have their proofs written out, which would make the entire endeavor a bit verbose. Still, I think an important part of proof-writing is looking for insights that arise and using those to refine one’s work.

In the case of your Theorem, the result follows because you end up with an integer that is a multiple of 2; well, where did that 2 come from? Etc.

]]>Let x be an element of Zmod4. Let Y be x + 4. Note that y is equivalent to x.

I think ‘equivalent’ is the correct word to describe any relation that is symmetrical, transitive and reflexive.

]]>In many way a mind map or conceptual map of non-physical entities can be converted into a spatial map. We already do this when we ask questions like how many years will it be before technology x enables us to do task y?

I’ve actually created product roadmaps that leverage both temporal and spatial relationships. These become topographic maps where distant events are rendered as topological maps.

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