Another Note on Paraconsistent Neutrosophic Sets
Abstract
:1. Introduction
2. Materials and Methods
- sup T = tsup inf T = tinf,
- sup I = isup, inf I = iinf;
- sup F =fsup , inf F = finf and
- nsup = tsup + isup + fsup , ninf = tinf + iinf + finf.
- (01)
- 0N = x (0, 0, 1)
- (02)
- 0N = x (0, 1, 1)
- (03)
- 0N = x (0, 1, 0)
- (04)
- 0N = x (0, 0, 0)
- (11)
- 1N = x (1, 0, 0)
- (12)
- 1N = x (1, 0, 1)
- (13)
- 1N = x (1, 1, 0)
- (14)
- 1N = x (1, 1, 1)
- (I1)
- A∩B = x (TA⋅TB, IA⋅IB, FA⋅FB)
- (I2)
- A∩B = x (TA∧TB, IA∧IB, FA∨FB)
- (I3)
- A∩B = x (TA∧TB, IA∨IB, FA∨FB)
- (U1)
- A∪B = x (TA∨TB, IA∨IB, FA∧FB)
- (U2)
- A∪B = x (TA∨TB, IA∧IB, FA∧FB)
- (1)
- ∩Aj may be defined as:
- (i)
- ∩Aj = x (∧, ∧, ∨)
- (ii)
- ∩Aj =x (∧, ∨, ∨)
- (2)
- ∪Aj may be defined as:
- (i)
- ∪Aj = x (∨, ∨, ∧)
- (ii)
- ∪Aj = x (∨, ∧, ∧)
- (1)
- 0N and 1N ϵ ττ;
- (2)
- G1∩G2 ∈ τ, for any G1;, G2 ∈ τ;
- (3)
- ∪Gj ∈ τ or any subfamily {Gj}j ∈ J of τ.
3. Results
- (1)
- It is necessary to omit a definition of ∩, because we will need ∩ of paraconsistent NSs to be paraconsistent. Indeed, let A = x (1/2, 1/2, 1/2) and B = x (1/2, 1/3, 1/3) (both are paraconsistent NSs), but 1/4 + 1/6 + 1/6 is not > 1. Then, the case with product ((I1), in Definition 4) must be deleted for paraconsistent NSs.
- (2)
- The definitions of 0N and 1N also have problems for paraconsistent NSs:
- (a)
- Only (02) and (12), (13), (1₄) are paraconsistent;
- (b)
- If we want all NSs: 0N∪0N, 0N∪1N, 1N∪1N, 0N∩0N, and 0N∩1N to be paraconsistent NSs, it is necessary to delete 12 in Definition 3, because with this definition,
- (1)
- 0N = x (0,1,1), and 1N = x (1,1,0) or x (1,1,1), are in τ;
- (2)
- G1∩G2 ∈ τ for any G1, G2 ∈ τ (where ∩ is defined by (I2) or (I3));
- (3)
- ∪Gj ∈ τ for any subfamily {Gj}j ∈ J of τ (where ∪ is defined by Definition 5).
4. Discussion
- (a)
- 0N = x (0,1,1), and 1N = x (1,1,0) or x (1,1,1)
- (b)
- ∩ defined by (I2) or (I3)
- (c)
- ∪ defined by Definition 5 is a bounded lattice.
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Lupiáñez, F.G. Another Note on Paraconsistent Neutrosophic Sets. Symmetry 2017, 9, 140. https://doi.org/10.3390/sym9080140
Lupiáñez FG. Another Note on Paraconsistent Neutrosophic Sets. Symmetry. 2017; 9(8):140. https://doi.org/10.3390/sym9080140
Chicago/Turabian StyleLupiáñez, Francisco Gallego. 2017. "Another Note on Paraconsistent Neutrosophic Sets" Symmetry 9, no. 8: 140. https://doi.org/10.3390/sym9080140
APA StyleLupiáñez, F. G. (2017). Another Note on Paraconsistent Neutrosophic Sets. Symmetry, 9(8), 140. https://doi.org/10.3390/sym9080140