By Karl Fink

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**Additional info for A brief history of mathematics;: An authorized translation of Dr. Karl Fink's Geschichte der Elementar-Mathematik,**

**Example text**

Fm ) (X ∗ ) = ψ(f1 (X ∗ ), f2 (X ∗ ), . . 2) simply boils down to function composition). 3 Duality 13 Boolean functions on B n , then the functions f ∨ g, f ∧ g, and f are defined, for all X∗ ∈ Bn , by (f ∨ g)(X∗ ) = f (X ∗ ) ∨ g(X∗ ), (f ∧ g)(X∗ ) = f (X ∗ ) ∧ g(X∗ ), f (X ∗ ) = f (X ∗ ). 8. The dual of a Boolean function f is the function f d defined by f d (X) = f (X) for all X = (x1 , x2 , . . , xn ) ∈ Bn , where X = (x 1 , x 2 , . . , x n ). 6. Let f be the 2-variable function defined by f (0, 0) = f (0, 1) = f (1, 1) = 1 and f (1, 0) = 0.

5, in particular, may initially seem confusing, since it equates a function with a formal expression, but this notational convention is actually innocuous: It is akin to the convention for real-valued functions of real variables, where it is usual to assimilate a function with its analytical expression and to write, for instance, equalities like f (x, y) = x 2 + 2xy + y 2 = (x + y)2 . 1)), it also naturally leads to the next notion. 6. We say that two Boolean expressions ψ and φ are equivalent if they represent the same Boolean function.

Let f be a Boolean function on Bn , and let k ∈ {1, 2, . . , n}. Then, f (x1 , x2 , . . 15) for all (x1 , x2 , . . , xn ) ∈ Bn . Proof. 15). 15) is often called the Shannon expansion of the function f with respect to xk , by reference to its use by Shannon in [827], although this identity was already well-known to Boole [103]. 11). More interestingly, by applying the Shannon expansion to a function and to its successive restrictions until these restrictions become either 0, or 1, or a literal, we obtain an orthogonal DNF of the function (this is easily proved by induction on n).

### A brief history of mathematics;: An authorized translation of Dr. Karl Fink's Geschichte der Elementar-Mathematik, by Karl Fink

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