Download e-book for kindle: An introduction to crystal analysis by William Henry Bragg

By William Henry Bragg

This ebook is based on a process lectures which was once given on the college university, Aberystwyth, within the autumn of 1926. The lectures have been meant for a common viewers ; and consisted, for that reason, of a sequence of easy illustrations of the most beneficial properties of crvstal research. it is going to now not were attainable to do extra in the limits imposed via time and the conditions. The remedy during this publication is a bit of fuller and extra different than that of the lectures.

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4. I(J-l(v)) = v /\ If(X) 5. 1- 1 (Cv) = C(J-l(v)). ~ v. if and only if I is injective. 7. Equality holds universally in 4 if and only if I is surjective. 8. I (Cp,) ~ Cu (p,)) holds universally if and only if I is injective. 9. I (Cp,) ~ Cu (p,)) holds universally if and only if I is surjective. 10. lf p, = c is constant, then I(p,) = c /\ 1 f(X). 11. lf v = c is constant, then 1-1 (V) = c. 6. Equality holds universally in 3 Proof. All properties follow easily from the definitions. 4 Proposition lf X and Y are sets, I:X-Y is afunction, and arefamilies offuzzy sets on respectively X and Y, (/lj)jEJ' (Vj)jEJ then: 1.

The following result puts this in its correct perspective. 6 Proposition Suppose given sets Xl, ... , X n and Y and a function f : Xl X ... X Xn ----+ Y. Then the function obtained by the extension principle is the image ofthe product fLl X ... X fLn under f. 0 The main use of the extension principle consists in the possibility of extending operations on the points of a product space to the product of the sets of fuzzy subsets, as can be seen in the chapter dealing with fuzzy real numbers, where addition is defined by means of this technique.

Proof. By straightforward verification. 6 Notes Fuzzy sets Several other, either more general or slightly different definitions of fuzzy sets have been given in the literature. Thus in Goguen in [JMAA 18, 1967, 145-174] we find the concept of L-fuzzy set, where L is a lattice. Given an arbitrary set X, an L-fuzzy set (on X) is a function from X to the lattice L, JL : X --+ L. Operations Although infimum and supremum are the most widely used extensions of interseetion and union, a host of other extensions have been considered in the literature.

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