By Robert A. Doughty

ISBN-10: 0208020969

ISBN-13: 9780208020963

An exam of the army doctrine that lively the French safeguard opposed to the German invasion in 1940.

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**Additional info for The seeds of disaster: The development of French army doctrine, 1919-1939**

**Example text**

1. 3 Fuzzy Set-valued Random Elements 39 (ii) The support {x : A( x) ! 0} of A is bounded. (iii) The D -level set AD of A are closed intervals From the above definition, it is not hard to show that? (a) Real numbers are fuzzy numbers. e. AD is closed, D >0,1@ . e. intervals). As stated above, the use of intervals and their arithmetic is appropriate in some situations involving imprecision. When the intervals themselves are not sharply defined, we arrive at the concept of fuzzy intervals. e. x R such that A( x) 1 ).

3 Fuzzy Relations 27 For negation operators, we can also consider the class of operators K : [0,1] o [0,1] , such that: (a) K (0) 1, K (1) 0 (b) K is non-increasing (c) K ( K (x)) = x similarly, fuzzy implication operators are generalizations of material implication operator of two-valued logic. Specifically, a fuzzy implication is a map : [0,1] u [0,1] o [0,1] , where restriction to {0,1} u {0,1} coincides with the truth table of the material implication. Some examples are 1 if x d y (i) ( x y ) = ® ¯0 if x !

Such a partition is referred to as a coarsening of X. Providing a model for observing X. If we consider the set-valued map S(w) = the element of the partition containing X(w), then S is formally a random set with probability density P( S Ai ) = P ( X Ai ) , i=1,…,k Thus, coarse data are actually observations of random sets which are special cases of random fuzzy sets. The following is an example of statistical estimation in the context of coarse data. For details, see [38]. Suppose the random variable X admits a probability density function f ( x | T ) , where T 4 R .

### The seeds of disaster: The development of French army doctrine, 1919-1939 by Robert A. Doughty

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