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Table Of Contents
Mathematics of Machine Learning
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The building blocks of mathematics are propositions of the form A →B; at least, this is what I emphasized throughout this chapter.
I was not precise. The proposition A →B translates to “if A, then B,” but sometimes, we know much more. Quite frequently, A and B have the same truth values. In natural language, we express this by saying “A if and only if B.” (Although this is much rarer than the simple conditional.)
In logic, we express this relation with the biconditional connective ↔︎, defined by
Theorems of the “if and only if” type are called equivalences, and they play an essential role in mathematics. When proving an equivalence, we must show both A →B and B →A.
To see an example, let’s go back to elementary geometry. As you probably learned in high school, we can describe geometric objects on the plane with vectors that are represented by a tuple of two real numbers. This...