Pythagorean fuzzy sets are an extension of intuitionistic fuzzy sets and are more efficient from an application perspective. Though the Pythagorean fuzzy sets are more informative, not much work on similarity measures is available in the literature. Furthermore, existing similarity measures are not efficient. Also, the containment property in Pythagorean fuzzy units is not correctly defined or ineffective. As a result, the existing similarity measures do not reflect appropriate information between the Pythagorean fuzzy sets. The scalar function of the Yager is mainly used for verifying the validity of similarity measures. Most of the existing similarity measures do not conform to the Yager scalar function. Hence, the existing similarity measures exhibit some discrepancies. Furthermore, the existing similarity measures are inconsistent in determining the similarity in intuitionsitic and Pythagorean fuzzy sets. In some real-world modeling issues, past, present, and cross-time information are essential. However, such information is missing in the existing similarity measures. Therefore, in this paper, two new measures of similarity are being developed based on the deviation of the parameters: membership degree, nonmembership degree, strength of commitment, direction of commitment, and cross-time evaluation factors. Under this construction, the proposed similarity measures effectively measure the similarity between the Pythagorean fuzzy sets. Furthermore, the newly defined containment property is also reflected in the proposed similarity measures, which were a limitation in most cases. Moreover, Yager's scalar function is also reflected by the proposed similarity measures. The complement of given information is also essential in some real-world problems. However, such information is incomplete in the theory of Pythagorean fuzzy sets. Hence, the complement of the Pythagorean fuzzy set is being redefined, and a few related results on similarity measures are proposed. Finally, the proposed similarity measures are tested for applicability to medical diagnosis and clustering problems through some hypothetical case studies.

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毕达哥拉斯模糊集的离散相似性测度及其在医学诊断和聚类问题中的应用

毕达哥拉斯模糊集是直觉模糊集的扩展，从应用的角度来看更有效。尽管毕达哥拉斯模糊集提供的信息更多，但文献中没有太多关于相似性度量的工作。此外，现有的相似性措施效率不高。此外，毕达哥拉斯模糊单元中的包含属性未正确定义或无效。因此，现有的相似性度量不能反映毕达哥拉斯模糊集之间的适当信息。Yager的标量函数主要用于验证相似性度量的有效性。大多数现有的相似性度量都不符合 Yager 标量函数。因此，现有的相似性度量表现出一些差异。此外，现有的相似性度量在确定直觉和毕达哥拉斯模糊集的相似性时不一致。在一些现实世界的建模问题中，过去、现在和跨时间信息是必不可少的。但是，现有的相似性度量中缺少此类信息。因此，在本文中，基于参数的偏差开发了两种新的相似性度量：成员度、非成员度、承诺强度、承诺方向和跨时间评估因素。在此结构下，所提出的相似性度量有效地测量了毕达哥拉斯模糊集之间的相似性。此外，新定义的包含属性也反映在提议的相似性度量中，这在大多数情况下是一个限制。此外，雅格 s 标量函数也反映在所提出的相似性度量中。给定信息的补充在一些现实世界的问题中也是必不可少的。然而，这些信息在毕达哥拉斯模糊集理论中是不完整的。因此，重新定义了毕达哥拉斯模糊集的补集，并提出了一些关于相似性度量的相关结果。最后，通过一些假设的案例研究，测试了所提出的相似性度量对医学诊断和聚类问题的适用性。并提出了一些关于相似性度量的相关结果。最后，通过一些假设的案例研究，测试了所提出的相似性度量对医学诊断和聚类问题的适用性。并提出了一些关于相似性度量的相关结果。最后，通过一些假设的案例研究，测试了所提出的相似性度量对医学诊断和聚类问题的适用性。