Exercise 1: Why the Two Sentences Produce Mathematically Identical Vectors — Possible Solution ==================================================================== THE VOCABULARY AND VECTORS, PER THIS CHAPTER ------------------------------ Per this chapter, with vocabulary [bites, dog, man]: "dog bites man" → [1, 1, 1] "man bites dog" → [1, 1, 1] WHY BAG-OF-WORDS COUNTING PRODUCES THIS RESULT ------------------------------ Per nlp1-2's own definition, a bag-of-words vector's value at each position is simply "how many times that word appears in that document" — a pure count operation with no reference anywhere to WHERE in the sentence a word occurs. Both "dog bites man" and "man bites dog" contain the word "bites" exactly once, the word "dog" exactly once, and the word "man" exactly once — the counting operation produces the value 1 at each of the three vocabulary positions for BOTH sentences, because counting occurrences is blind to sequence position by its own definition. WHY TF-IDF DOESN'T CHANGE THIS EITHER ------------------------------ Per nlp1-3, TF-IDF's own formula (TF × IDF) operates on exactly these same starting counts — TF is a rate derived from the count, and IDF depends only on how many DOCUMENTS (not positions within a document) contain a given word across the whole corpus. Since both sentences share identical starting counts (as shown above) and would be evaluated against the identical corpus-wide document statistics, every step of the TF-IDF calculation produces identical results for both sentences — there's no step anywhere in the formula where word position could enter the calculation. WHY THIS IS IDENTICAL, NOT MERELY SIMILAR ------------------------------ This isn't a case of the two vectors happening to end up close to each other by approximation — both representations are built from a mechanical counting/weighting process that has literally no input variable representing position at all. Two different arrangements of the exact same three words, appearing exactly once each, cannot produce two different results from a process that never once looks at arrangement — the vectors are exactly, not approximately, equal, down to every digit, because the underlying calculation genuinely has no way to distinguish the two inputs from each other. WHY THIS PROVES RATHER THAN MERELY ASSERTS THE LIMITATION ------------------------------ Working through the actual vector values for both sentences, using the chapter's own defined vocabulary and both chapters' own counting/ weighting rules, demonstrates the claim directly and verifiably rather than simply stating "bag-of-words loses word order" as an assumption to be taken on faith — the identical output for two genuinely different sentences is concrete, checkable evidence of the limitation. WHY THIS WORKS AS AN ANSWER ------------------------------ It works through the counting mechanism for both sentences explicitly, using the chapter's own defined vocabulary, to show why both nlp1-2's counts and nlp1-3's TF-IDF weights are mathematically identical for both sentences — a direct, verifiable proof rather than a restated assertion.