The Ultimate Outline Of SCH900776

Definition 3 (see [6]) ��Let A LX and B LY. Define an L-fuzzy set A �� B on X �� Y by(A��B)(x,y)=A(x)��B(y),??(x,y)��X��Y.(two)A �� B is called the merchandise new product of the and B.Definition four (see [6]) ��Let A LX, B LY, andR LX��Y. An L-fuzzy relation f from A to B is called an L-fuzzy mapping from (of) A into B if a M(L), f[a] can be a mapping from (of) A into B, after which we compose it as f : A �� B.Theorem six (see [7]) ��Let A LX, B LY, and f �� A �� B, then the following ailments are equivalent:f is an L-fuzzy mapping from A into B;for each a M(L), f(a) is often a mappingnamely from A(a) into B(a);for every a P(L), f(a) can be a mapping from A(a) into B(a).Theorem 7 (see [8]) ��Let A LX, B LY, and f �� A �� B.

If for every b, c L, ��*(bc) = ��*(b)��*(c), then the following circumstances are equivalent:f is definitely an L-fuzzy mapping from A into B;for every a ��*(0), f[a] is usually a mapping from A[a] into B[a].Definition 8 (see [9]) ��Let A LX, B LY, and f : A �� B. For C �� A, we define f(C) = aM(L)(a��f[a](C[a])). Then f(C) is named the picture of C underneath f.Theorem 9 (see [9]) ��Let A LX, B LY, and f : A �� B, and then for C �� A we havefor every a M(L), f(C)(a)f(a)(C(a))f[a](C[a])f(C)[a],f(C) = aM(L)(a��f(a)(C(a))),for each a P(L), f(C)(a) = f(a)(C(a)),f(C) = aP(L)(af(a)(C(a))). Definition 10 (see [9]) ��Let A LX, B LY, and f : A �� B. For D �� B, we define f?one(D) = aM(L)(a��(f[a])?one(D[a])). Then f?one(D) is known as theBay 11-7085 inverse image of D under f.Theorem eleven (see [9]) ��Let A LX, B LY, and f : A �� B.

Then for D �� B, we havefor each a M(L), (f?1(D))(a)(f(a))?1(D(a))(f[a])?one(D[a])(f?1(D))[a],f?1(D) = aM(L)(a��(f(a))?1(D(a))),for every a P(L), (f?1(D))(a) = (f(a))?one(D(a)),f?one(D) = aP(L)(a(f(a))?1(D(a))). Definition twelve (see [10]) ��Let X be a set, A LX, along with a �� 0. An L-fuzzy relation R from A to A is called an L-fuzzy partial order on the if R satisfies the following circumstances:for all x A(0), R(x, x) = A(x),RR �� R,for all x, y A(0), R(x, y)��R(y, x) �� 0x = y.When R is an L-fuzzy partial purchase on a, we phone (A, R) an L-fuzzy partial purchase set or L-poset for quick.Theorem 13 (see [10]) ��For an L-fuzzy relation R on the, the next implications (four)(1) and (1)(two)(three)(five)(6)(7) are real.R is surely an L-fuzzy partial purchase on the.For every a L0, if R[a] will not be empty set, then it really is a partial purchase on A[a].

For every single a M(L), if R[a] is just not empty set, then it can be a partial order on A[a].For each a ��(1), if R(a) is not really empty set, then it's a partial order on the(a).For every a ��(0), if R[a] is just not empty set, then it really is a partial buy on A[a].For each a ��*(0), if R[a] is not empty set, then it can be a partial purchase on A[a].For every a P(L), if R(a) is not really empty set, then it really is a partial purchase on the(a).Remark 14 (see [10]) ��In basic, (1)(four) in the previous theorem will not be genuine.