@MISC{FUZNUM_1.MIZ,
  AUTHOR = {Grabowski, Adam},
  TITLE = {The Formal Construction of Fuzzy Numbers},
  SECTION1 = {Preliminaries: Affine Maps},
  SECTION2 = {Towards Development of Fuzzy Numbers},
  SECTION3 = {Convexity and the Height of a Fuzzy Set},
  SECTION4 = {Pasting aka Glueing Lemmas},
  SECTION5 = {Triangular and Trapezoidal Fuzzy Sets},
  DAY = {31},
  MONTH = {December},
  YEAR = {2014},
  ADDRESS1 = {Institute of Mathematics\\ University of Bialystok\\ ul. Akademicka 2\\ 15-267 Bia\l ystok, Poland},
  SUMMARY = {In this article, we continue the development of the theory of fuzzy sets
  \cite{Zadeh:1965}, started
 with \cite{FUZZY_1.ABS} with the future aim to provide the formalization
 of fuzzy numbers \cite{DuboisPrade:1978} in terms reflecting the current state of the
 Mizar Mathematical Library. Note that in order to have more usable
 approach in \cite{FUZZY_1.ABS}, we revised that article also.
 As we can actually understand fuzzy sets just as their membership functions
 (via the equality of membership function and their set-theoretic
 counterpart), all the calculations are much simpler.
 To test our simple approach,
 we give the notions of (normal) triangular and trapezoidal fuzzy sets as the
 examples of concrete objects. Also $\alpha$-cuts, the core of a fuzzy set,
 and normalized fuzzy sets were defined.
 Main technical obstacle was to prove continuity of the glued maps,
 and in fact we did this not through its topological counterpart,
 but extensively reusing properties of the real line (with loss of
 generality of the approach, though), because we aim at formalizing
 fuzzy numbers in our future submissions, as well as merging with
 rough set approach as introduced in \cite{ROUGHS_1.ABS}.
 Our base for formalization was \cite{DuboisPrade:1980}.}}

@BOOK{DuboisPrade:1980,
 AUTHOR={Dubois, Didier and Prade, Henri},
 TITLE={Fuzzy Sets and Systems: Theory and Applications},
 PUBLISHER={Academic Press, New York},
 YEAR=1980}

@ARTICLE{Zadeh:1965,
 AUTHOR={Zadeh, Lotfi},
 TITLE={Fuzzy sets},
 JOURNAL={Information and Control},
 YEAR=1965,
 VOLUME=8,
 NUMBER=3,
 PAGES={338--353}}

@ARTICLE{DuboisPrade:1978,
 AUTHOR={Dubois, Didier and Prade, Henri},
 TITLE={Operations on fuzzy numbers},
 JOURNAL={International Journal of System Sciences},
 YEAR=1978,
 VOLUME=9,
 NUMBER=6,
 PAGES={613--626}}

@ARTICLE{DuboisPrade:1990,
 AUTHOR={Dubois, Didier and Prade, Henri},
 TITLE={Rough fuzzy sets and fuzzy rough sets},
 JOURNAL={International Journal of General Systems},
 YEAR=1990,
 VOLUME=17,
 NUMBER=2--3,
 PAGES={191--209}}

