About the Images and Inverse Images of Intuitionistic or Vague Fuzzy Subsets

In this paper an exclusive study of some standard (lattice) algebraic properties for Intuitionistic fuzzy (inverse) images of Intuitionistic fuzzy subsets is done. Further as in crisp setup, characterizations for injectivity and surjectivity of maps in terms of some (lattice) algebraic properties of Intuitionistic fuzzy images and Intuitionistic fuzzy inverse images are performed.


Introduction
The traditional view in science, especially in mathematics, is to avoid uncertainty at all levels at any cost. Thus, 'being uncertain' is regarded as 'being unscientific'. But unfortunately in real life most of the information that we have to deal with is mostly uncertain. One of the paradigm shifts in science and mathematics in this century is to accept uncertainty as part of science and the desire to be able to deal with it, as there is very little left out in the practical real world for scientific and mathematical processing without this acceptance! One of the earliest successful attempts in this direction is the development of the Theories of Probability and Statistics. However, both of them have their own natural limitations. Another successful attempt again in the same direction is the so called Fuzzy Set Theory, introduced by Lotfi Zadeh.
According to Zadeh, a fuzzy Subset of a set X is a function μ from X to the closed interval [0,1] of real numbers. The function μ, he called, the membership function which assigns to each member x of X its membership value, μx in [0, 1].
Fuzzy set theory is one area with large number of applications both in Mathematics and in Computer science. For applications of fuzzy set theory in Mathematics, one can refer to the text books Mordeson and Malik [10] in Fuzzy Algebra and Ying-Ming and Mao-Kang [15] in Fuzzy Topology. For applications of Fuzzy set theory in Computer science, one can refer to the text books Galindo-Urrutia and Piattini [5] in Fuzzy Data Base Management Systems, Tamalika and Ajoy [12] in Fuzzy Image Processing, Hiroshi [7] and Morgan [9] in Fuzzy Data Mining, Zhang [14] and Liu and Li [11] in Fuzzy Neural Networks, Valente and Pedrycz [8] in fuzzy clustering. In 1983, Atanassov [1] generalized the notion of Zadeh fuzzy subset of a set further by introducing an additional function , which he called a non-membership function with some natural conditions on μ and , calling these new generalized fuzzy subsets of a set, Intuitionistic fuzzy subsets. Thus according to him an Intuitionistic fuzzy subset of a set X, is a pair A=(μ A ,  A ), where μ A ,  A : X→ [0, 1] of real numbers such that for each x in X, μ(x) +  (x) ≤ 1, where μ A is called the membership function of A and μ A is called the non -membership function of A. Interestingly the same notion of Intuitionistic fuzzy subset of a set was also introduced by Gau and Buehrer [6] in 1993 under a different name called Vague subset. Thus whether we called Intuitionistic fuzzy subset of a set or if-subset of a set for short, or vague subset of a set, they are one and the same. In stead of using long phrases like Intuitionistic fuzzy subset or vague subset, here onwards, we use the phrase if-subset. Obviously, if/v-subset only means Intuitionistic fuzzy/vague subset. Ever since Atanassov [1] introduced the notion of Intuitionistic fuzzy subset of a set, several mathematicians started imposing and studying both algebraic and topological structures on Intuitionistic fuzzy subsets. Looking at several of the papers that are in print and online, one thing which becomes evident is that various (lattice) algebraic properties of images and inverse images of a Intuitionistic fuzzy subset which, incidentally, not only play a crucial role in the study of both Intuitionistic Fuzzy Algebra and Intuitionistic Fuzzy Topology but also are necessary for the individual/ exclusive development of Intuitionistic Fuzzy Set Theory, are not yet studied, although these concepts were existing since long. In fact, these concepts of if/v-image and if/v inverse image are dealt in Ming [17], Thakur and Pandey [13], Davvaz, Dudek and Jun [4], Yon, Jun and Kim [16]. However, in this paper, we make an exclusive and somewhat detailed study of these (lattice) algebraic properties of Intuitionistic fuzzy images and Intuitionistic fuzzy inverse images under crisp maps. Further as in crisp setup, we characterize some injectivity and surjectivity of maps in terms of some (lattice) algebraic properties of Intuitionistic fuzzy images and Intuitionistic fuzzy inverse images. A few of the results in this paper may be available in the literature elsewhere but scattered; however, we presented them here not only collectively but also in a suitable way for further research to the individual/exclusive development of Intuitionistic fuzzy Set Theory.

Main Results
In this section the notions and properties of Intuitionistic fuzzy/vague image and Intuitionistic fuzzy/ vague inverse image for an Intuitionistic fuzzy/vague subset of a set under a crisp map are recalled and are shown to be well defined.

if-Images and if-Inverse Images
Let X, Y be a pair of sets and let f: X → Y be a map. Let A = (μ A ,  A ) and B= (μ B ,  B ) be if/vsubsets of X and Y respectively. (4) Let X, Y, f and B be as above. Then the if/v-subset C of X defined as in (3)

Mapping Properties of if-Images and if-Inverse Images
In this section we show that several of the mapping properties that hold good for Zadeh fuzzy subsets are also held good for the Intuitionistic fuzzy subsets.

Theorem
Let X, Y be a pair of sets and let f:X→Y be a map. Let A, Ai and B, B i be if-subsets of X and Y respectively. Then the following are true:

Theorem
Let X, Y and Z be three sets and let f: X→Y and g: Y→Z be a pair of maps. Then the following are true: