Welcome back to the nth instalment (because I am not counting how many times I have written) of blog posts concerning mathematical expressions and logic! Due to Thanksgiving, we only had two hours of class this past week, so we dealt with proofs that do not involve boolean values, disproving, cases, and delta-epsilon proofs.
Concerning non-boolean proofs, the definition of floor was utilized. In English, the definition is: for all real numbers, the floor of a real number is the same as the integer that is less than or equal to the specified real number and this integer has no other integers greater than it that are closer to the specified real number. It is the same thing as dividing any real number and only looking at the whole number value of it. In Python, this is signified by "//", so 5 // 3 would return 1.
I never thought that mixed numbers would reappear after having only used them up to grade five in elementary school.
For proving things false, it was similar to the typical idea of proving something wrong by providing a counterexample (in the case of Universal statements), but with structure.
As for proofs by cases, this is inception, but with proofs: proof-ception, where in order to prove or disprove a statement, one must prove the statement true/false for a specific category, number, and the like, and then do the same proof for the next related category. It is ironic, that in order to prove something as simple as all natural numbers squared added to themselves will be even, we must build an extensive and relatively complex proof for it, in spite of the fact that academia likes to assert that--if we truly understand something--we can reduce it to its simplest form with relative ease.
Is a fourteen line proof for a simplistic concept support this idea that understanding implies a simple explanation? I believe it disproves this implication.
Sure, it may be true in a number of skills and fields, but it most certainly does not seem to be a universally true statement.
Thanks for
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