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In this section, we generalize the definitions and results of intuitionistic fuzzy frontier in the intuitionistic fuzzy semisettings. In the following theorems, we note that almost all the properties related to intuitionistic fuzzy semi-interior, intuitionistic fuzzy semi-closure and intuitionistic fuzzy semifrontier are analogous to their counterparts in Intuitionistic Fuzzy Topology, and hence proofs of most of them are not given.
This gives the equality. In the following theorem, 1 — 5 are analogues of Theorem 12 , and hence we omit their proofs. The converse of 2 , 3 , 6 , and 7 of Theorem 44 is, in general, not true as is shown by the following. The following is an analogue of Theorem To show that 2 , 3 , and 4 of Theorem 46 , are, in general, irreversible, in Example 3.
Whereas, in Intuitionistic Fuzzy Topology, we give counter-examples to show that these may not hold in general. The converse of Theorem 50 is, in general, not true as is shown by the following. However, we have the following theorem which is an analogue of Theorem The analogue of Theorem 22 is the following theorem, the proof of which is easy to establish. As in the case of Theorem 22 2 we also do not know whether the equality in Theorem 54 2 holds or not. So we leave these as open problems. The reverse inclusion is obvious. Thus, f is intuitionistic fuzzy semi-continuous.
Then one has. Suppose that f is intuitionistic fuzzy semi-continuous. The author declares that there is no conflict of interests regarding the publication of this paper. National Center for Biotechnology Information , U. Journal List ScientificWorldJournal v. Published online Feb Author information Article notes Copyright and License information Disclaimer. Received Aug 29; Accepted Oct This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract Notions of frontier and semifrontier in intuitionistic fuzzy topology have been studied and several of their properties, characterizations, and examples established. Preliminaries Definition 1 see [ 16 ] — Let X be a nonempty fixed set. Example 15 — To show that 2 , 3 , and 4 of Theorem 14 are, in general, irreversible, we choose in Example 3. Example 17 — In Example 3. Example 19 — In Example 3. Example 21 — To show that the converse of Theorem 20 is in general not true, in Example 3.
Remark 23 — We checked 2 of Theorem 22 on a large number of IFTSs, no counter-example could be found to establish the irreversibility of inequality. Example 24 — In Example 3. Proof — Suppose that f is IF continuous. Intuitionistic Fuzzy Semifrontier Levine [ 20 ] generalized the notion of open sets as semiopen sets.
Remark 42 — In the following theorems, we note that almost all the properties related to intuitionistic fuzzy semi-interior, intuitionistic fuzzy semi-closure and intuitionistic fuzzy semifrontier are analogous to their counterparts in Intuitionistic Fuzzy Topology, and hence proofs of most of them are not given. Example 45 — In Example 3. Example 47 — To show that 2 , 3 , and 4 of Theorem 46 , are, in general, irreversible, in Example 3. Example 49 — In Example 3.
Example 51 — In Example 3.
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Remark 55 — As in the case of Theorem 22 2 we also do not know whether the equality in Theorem 54 2 holds or not. Example 56 — In Example 3. Proof — Suppose that f is intuitionistic fuzzy semi-continuous. Conflict of Interests The author declares that there is no conflict of interests regarding the publication of this paper.
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