By Zhaohui Luo

This booklet develops a sort thought, stories its homes, and explains its makes use of in laptop technological know-how. The ebook focuses specifically on how the examine of kind conception could supply a robust and uniform language for programming, application specification and improvement, and logical reasoning. the kind idea built the following displays a conceptual contrast among logical propositions and computational information varieties. ranging from an advent of the fundamental options, the writer explains the that means and use of the type-theoretic language with proof-theoretic justifications, and discusses numerous matters within the research of sort conception. the sensible use of the language is illustrated by means of constructing an method of specification and information refinement in sort thought, which helps modular improvement of specification, courses, and proofs. scholars and researchers in machine technology and good judgment will welcome this fascinating new ebook.

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45 46 CHAPTER 3. FUZZY QUANTITIES This means that (A ◦ B)(x) = = _ ∧(A × B) ◦−1 (x) _ ∧(A × B)(a, b) a◦b=x = _ {A(a) ∧ B(b)} a◦b=x For example, for the ordinary arithmetic binary operations of addition and multiplication on R, we then have corresponding operations A + B = ∨ ∧ (A × B)+−1 and A · B = ∨ ∧ (A × B)·−1 on F(R). Thus (A + B)(z) = W x+y=z (A · B)(z) = W x·y=z {A(x) ∧ B(y)} {A(x) ∧ B(y)} The mapping R → R : r → −r induces a mapping F(R) →F(R) and the image of A is denoted −A. For x ∈ R, W (−A)(x) = x=−y {A(y)} = A(−x) If we view − as a binary operation on R, we get W {A(x) ∧ B(y)} (A − B)(z) = x−y=z It turns out that A + (−B) = A − B, as is the case for R itself.

Let f : U → V , g : V → W , and h : W → X. Show that h(gf ) = (hg)f . That is, show that composition of functions is associative. 30. For any sets X and Y, let M ap(X, Y ) be the set of all mappings from X to Y. Show that Φ : M ap(W, M ap(X, Y )) → M ap(W ×X, Y ) : Φ(f )(w, x) = f (w)(x) is a one-to-one correspondence. 31. Let f : U → V . (a) Show that f f −1 = 1P(V ) if f is onto. (b) Show that f −1 f = 1P(U ) if f is one-to-one. 32. Let f : U → V . Show that {f −1 f (x) : x ∈ U } is a partition of U .

Remember that ∧(A × B)(r, s) = A(r) ∧ B(s). The composition F(R) × F(R) → F(R × R) → F(R) of these two is the mapping that sends (A, B) to ∨(∧(A × B))◦−1 , where ◦−1 (x) = {(a, b) : a ◦ b = x}. We denote this binary operation by A ◦ B. 45 46 CHAPTER 3. FUZZY QUANTITIES This means that (A ◦ B)(x) = = _ ∧(A × B) ◦−1 (x) _ ∧(A × B)(a, b) a◦b=x = _ {A(a) ∧ B(b)} a◦b=x For example, for the ordinary arithmetic binary operations of addition and multiplication on R, we then have corresponding operations A + B = ∨ ∧ (A × B)+−1 and A · B = ∨ ∧ (A × B)·−1 on F(R).