Exercise 1: Why a Sigmoid Neuron Is Identical to ml1-5's Logistic Regression, Not Just Similar — Possible Solution
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THE TWO FORMULAS, SIDE BY SIDE
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Per this chapter:
  ml1-5's logistic regression: z = w1*x1 + w2*x2 + ... + wn*xn + b, output = sigmoid(z)
  a single neuron:             z = w1*x1 + w2*x2 + ... + wn*xn + b, output = activation(z)

WHY THIS IS EQUALITY, NOT RESEMBLANCE
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The two formulas share every single component: the same weighted-sum
structure (each input multiplied by its own weight, summed together),
the same bias term added afterward, and the same squashing step applied
to the result. The only symbol that differs at all between the two
formulas is the name given to the squashing function — "sigmoid" in
one, "activation" in the other. Once a neuron's own activation function
is specifically set to sigmoid, per this chapter's own tip-box, "the
formula is character-for-character identical to ml1-5's own logistic
regression." There is no additional step, no extra term, and no
different arithmetic operation anywhere in one formula that isn't
present, unchanged, in the other.

WHY "IDENTICAL" IS THE ACCURATE WORD, NOT "SIMILAR"
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Two things are "similar" when they resemble each other while remaining
distinct in some real respect. Here, substituting sigmoid for
"activation" doesn't produce something merely resembling ml1-5's own
formula — it produces the exact same sequence of arithmetic operations,
computed on the exact same kind of inputs, producing the exact same
kind of output (a value squashed into (0,1) via sigmoid, precisely as
ml1-5 already computed). There is no operation present in one version
that's absent, modified, or approximated in the other.

WHY THIS MEANS ml1-5's OWN TRAINED MODEL WAS ALREADY A NEURON
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Per this chapter's own tip-box, "you've already trained one of these —
ml1-5's own coefficients were literally a single neuron's own learned
weights." Since the formulas are identical when the activation is
sigmoid, the trained logistic regression model from ml1-5 — its own
specific learned weight and bias values — IS, without any translation,
reinterpretation, or adjustment, a single trained neuron. Nothing new
needed to be built or retrained to arrive at that neuron; it already
existed as ml1-5's own fitted model.

WHY THIS WORKS AS AN ANSWER
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It compares the two formulas term by term to show every component
matches exactly, explains why substituting "activation" with "sigmoid"
produces literal equality rather than mere resemblance, and connects
this directly to the chapter's own claim that ml1-5's own trained model
already was a single neuron, not something requiring separate
construction.