<abstract xmlns="http://www.w3.org/1999/xhtml">

<p>Since isosceles triangular and trapezoidal membership functions [<xref ref-type="bibr" rid="j_forma-2023-0006_ref_004">4</xref>] are easy to manage, they were applied to various fuzzy approximate reasoning [<xref ref-type="bibr" rid="j_forma-2023-0006_ref_010">10</xref>], [<xref ref-type="bibr" rid="j_forma-2023-0006_ref_013">13</xref>], [<xref ref-type="bibr" rid="j_forma-2023-0006_ref_014">14</xref>]. The centroids of isosceles triangular and trapezoidal membership functions are mentioned in this article [<xref ref-type="bibr" rid="j_forma-2023-0006_ref_016">16</xref>], [<xref ref-type="bibr" rid="j_forma-2023-0006_ref_009">9</xref>] and formalized in [<xref ref-type="bibr" rid="j_forma-2023-0006_ref_011">11</xref>] and [<xref ref-type="bibr" rid="j_forma-2023-0006_ref_012">12</xref>]. Some propositions of the composition mapping (<italic>f</italic> + · <italic>g</italic>, or f +* g using Mizar formalism, where <italic>f</italic>, <italic>g</italic> are a ne mappings), are proved following [<xref ref-type="bibr" rid="j_forma-2023-0006_ref_003">3</xref>], [<xref ref-type="bibr" rid="j_forma-2023-0006_ref_015">15</xref>]. Then different notations for the same isosceles triangular and trapezoidal membership function are formalized.</p>
<p>We proved the agreement of the same function expressed with different parameters and formalized those centroids with parameters. In addition, various properties of membership functions on intervals where the endpoints of the domain are fixed and on general intervals are formalized in Mizar [<xref ref-type="bibr" rid="j_forma-2023-0006_ref_001">1</xref>], [<xref ref-type="bibr" rid="j_forma-2023-0006_ref_002">2</xref>]. Our formal development contains also some numerical results which can be potentially useful to encode either fuzzy numbers [<xref ref-type="bibr" rid="j_forma-2023-0006_ref_007">7</xref>], or even fuzzy implications [<xref ref-type="bibr" rid="j_forma-2023-0006_ref_005">5</xref>], [<xref ref-type="bibr" rid="j_forma-2023-0006_ref_006">6</xref>] and extends the possibility of building hybrid rough-fuzzy approach in the future [<xref ref-type="bibr" rid="j_forma-2023-0006_ref_008">8</xref>].</p>
</abstract>

Since isosceles triangular and trapezoidal membership functions [4] are easy to manage, they were applied to various fuzzy approximate reasoning [10], [13], [14]. The centroids of isosceles triangular and trapezoidal membership functions are mentioned in this article [16], [9] and formalized in [11] and [12]. Some propositions of the composition mapping (f + · g, or f +* g using Mizar formalism, where f, g are a ne mappings), are proved following [3], [15]. Then different notations for the same isosceles triangular and trapezoidal membership function are formalized.
We proved the agreement of the same function expressed with different parameters and formalized those centroids with parameters. In addition, various properties of membership functions on intervals where the endpoints of the domain are fixed and on general intervals are formalized in Mizar [1], [2]. Our formal development contains also some numerical results which can be potentially useful to encode either fuzzy numbers [7], or even fuzzy implications [5], [6] and extends the possibility of building hybrid rough-fuzzy approach in the future [8].

Isosceles Triangular and Isosceles Trapezoidal Membership Functions Using Centroid Method

Formalized Mathematics

This work is licensed under the Creative Commons Attribution-ShareAlike 3.0 Public License.

Since isosceles triangular and trapezoidal membership functions [4] are easy to manage, they were applied to various fuzzy approximate reasoning [10], [13],...

Isosceles Triangular and Isosceles Trapezoidal Membership Functions Using Centroid Method

Pubblicato online: 26 set 2023

Pagine: 59 - 66

Accettato: 31 mar 2023

DOI: https://doi.org/10.2478/forma-2023-0006

Parole chiave
defuzzification, centroid method, isosceles triangular function, isosceles trapezoidal function

© 2023 Takashi Mitsuishi, published by Sciendo

This work is licensed under the Creative Commons Attribution-ShareAlike 3.0 Public License.

Isosceles Triangular and Isosceles Trapezoidal Membership Functions Using Centroid Method

Pubblicato online: 26 set 2023

Pagine: 59 - 66

Accettato: 31 mar 2023

DOI: https://doi.org/10.2478/forma-2023-0006

Parole chiavedefuzzification, centroid method, isosceles triangular function, isosceles trapezoidal function

© 2023 Takashi Mitsuishi, published by Sciendo

This work is licensed under the Creative Commons Attribution-ShareAlike 3.0 Public License.

Parole chiave
defuzzification, centroid method, isosceles triangular function, isosceles trapezoidal function