Before you can judge why one encoding is right and another is wrong, you need to be fluent in the little symbols the parent note throws at you: sets, subscripts, vectors, distance, the dot product, the Kronecker delta, and matrix rank. This page builds every one of them from zero, in the order they depend on each other.
Read that out loud: "the collection containing c1, c2, and so on up to cn."
The letter c (for category) is a placeholder — we don't know the actual word, so we use a letter.
The little number below, called a subscript, is a name tag that tells apart the first, second, third item. c1 = "first category", c2 = "second category".
The letter n is the count of how many categories exist. If we have {Red, Blue, Green}, then n=3.
Look at the figure: three unordered labels floating in a bag. Nothing says one comes before another. Our job is to give each a number-address while keeping that fairness.
The symbol ∈ means "is a member of". Picture an item sitting inside the bag. So i∈{0,1,2} says "i is one of 0, 1, or 2".
Zero-indexing means we start counting at 0, not 1. So n categories get the addresses 0,1,…,n−1. For n=3: the addresses are 0,1,2 — still three of them, just shifted to start at zero (a programming convention).
The parent uses two "less-than" signs and they mean different things.
The figure shows two number lines. Top (ordinal): shirt sizes land on 0,1,2,3 and the arrow of increasing number matches increasing real size — the order is earned. Bottom (nominal): countries dumped onto 0,1,2 — the arrow now lies, claiming Brazil is "twice" India.
One-hot encoding stops using a single number and starts using a row of numbers.
In the figure, three arrows point straight out along the three axes of a 3-D room. Each is a "one-hot" colour. Notice they are all the same length and sit at the same angle to each other — no colour is favoured. That symmetry is the whole reason one-hot exists.
Now the parent's line ei⋅ej=δij reads in plain words: "a one-hot arrow dotted with itself gives 1 (they fully agree), and dotted with a different one gives 0 (perfectly perpendicular)." Perpendicular arrows are the geometric picture of "totally independent, no accidental relationship."
Notice how the left branch (order symbols) feeds label encoding, while the right branch (vectors, distance, dot product) feeds one-hot. They meet at the final decision. Deeper consequences flow onward into Curse of Dimensionality (too many one-hot columns), Decision Trees (which handle label codes more gracefully), Target Encoding (a smarter alternative), and the broader craft of Feature Engineering.
