2.2. Generalized Fuzzy Semi-Pre-Open Sets (G_F SPOS)
Definition 2.2.1: Let (X,?) be G_F TS. A generalized fuzzy set G_F S ? in X is called generalized fuzzy semi-open (G_F SOS) if ??cl_? (i_? (?)).
Proposition 2.2.1: G_F OS(X) ? G_F SOS(X)
Proof: Let (X,?) be G_F TS and ? be G_F OS(X). Then i_? (?)=?. Since ??cl_? (?). This implies ?=i_? (?)?(cl_? (i_? (?) )). Hence ? is G_F SOS(X).
Remark 2.2.1: The converse of proposition 2.2.1 is not true as illustrated in Example 2.
2.1
Example 2.2.1: Let X={x_1,x_2}, A={?(x?_1,0.3),?(x?_2,0.7)}, B={?(x?_1,0.7),?(x?_2,0.4)} and C={?(x?_1,0.7),?(x?_2,0.7)}. Clearly ?={0,A,B,C.1} isGFT(X). Thus the set ?={?(x?_1,0.8),?(x?_2,0.7)} is GFSOS(X) but not GFOS(X).
Definition 2.2.2: Let (X,?) be GFTS. A GFS(X) ? is called generalized fuzzy pre-open (GFPOS) if ??i_? (cl_? (?)).
Proof: Let (X,?) be GFTS and ? be GFOS(X). Then cl_? (?)=?. Since ??i_? (?). This implies ?=cl_? (?)?(i_? (cl_? (?) )). Hence ? is GFPOS(X).
Remark 2.2.2: The converse of proposition 2.2.1 is not true as illustrated in Example 2.2.2
Example 2.2.2: In Example 3.1, ?={?(x?_1,0.8),?(x?_2,0.7)} is GFPOS(X)but not GFOS(X).
Definition 2.2.3: Let (X,?) be GFTS. A GFS(X) ? is called generalized fuzzy ?-open (GF?OS) if ??i_? (cl_? (i_? (?) )).
Proof: Let (X,?) be GFTS and ? be GFOS(X). Then i_? (?)=?. Since ??cl_? (?), we have ??cl_? (i_? (?)). This implies ?=i_? (?)?i_? (cl_? (i_? (?) )). Hence ? is GF?OS(X).
Remark 2.2.3: The converse of proposition 2.2.3 is not true as illustrated in Example 2.2.3
Example 2.2.3: In Example 2.
2.1, ?={?(x?_1,0.8),?(x?_2,0.7)} is GF?OS(X)but not GFOS(X).
Definition 2.2.4: Let (X,?) be GFTS. A GFS(X) ? is called generalized fuzzy ?-open (GF?OS) if ??cl_? (i_? (cl_? (?))).
Remark 2.2.4: The converse of proposition 2.2.4 is not true as illustrated in Example 2.2.4
Example 2.2.4: In Example 2.2.1, ?={?(x?_1,0.8),?(x?_2,0.7)} is GF?OS(X) but not GFOS(X).
Proposition 2.3.2: GFSOS(X) ? GFSPOS(X)
Proof: Let X be GFTS and let ?_1be GFSPOS(X) then ?_1?cl_? (i_? (?_1 ). Let us consider GFS, ?_2=i_? (?_1). Since ?_2 isGFOS(X)and a hence GFPOS(X). Now ?_2??_1?cl_? (?_2). This implies ?_1 is GFSPOS(X). Thus each GFSOS(X) is GFSPOS(X).
Remark 2.3.2: The converse of Proposition 2.3.2 is not true in general as shown in Example 2.3.2.
Example 2.3.2: In Example 2.3.1 ?_2 is GFSPOS(X) but not GFSOS(X)
Remember! This is just a sample.
You can get a custom paper by one of our expert writers.
Get your custom essay
Helping students since 2015
Essay Writing Service Features
Our Experience
No matter how complex your assignment is, we can find the right professional for your specific task. Contact Essay is an essay writing company that hires only the smartest minds to help you with your projects. Our expertise allows us to provide students with high-quality academic writing, editing & proofreading services.
Free Features
Free revision policy
$10Free bibliography & reference
$8Free title page
$8Free formatting
$8How Our Essay Writing Service Works
First, you will need to complete an order form. It's not difficult but, in case there is anything you find not to be clear, you may always call us so that we can guide you through it. On the order form, you will need to include some basic information concerning your order: subject, topic, number of pages, etc. We also encourage our clients to upload any relevant information or sources that will help.
Complete the order form
Once we have all the information and instructions that we need, we select the most suitable writer for your assignment. While everything seems to be clear, the writer, who has complete knowledge of the subject, may need clarification from you. It is at that point that you would receive a call or email from us.
Writer’s assignment
As soon as the writer has finished, it will be delivered both to the website and to your email address so that you will not miss it. If your deadline is close at hand, we will place a call to you to make sure that you receive the paper on time.
Completing the order and download