The notions of the bipolar complex fuzzy set (BCFS) and complex bipolar fuzzy set (CBFS) have been already given, but these notions of BCFS and CBFS have the problem that they contradict the basic definition of the complex numbers which we discussed in this article, and then we defined the new definition of BCFS. Our defined notion of BCFS is more closed to bipolarity as compared with already existing BCFS and CBFS, and more accurate. BCFS is the fusion of bipolar fuzzy set (BFS) which a decision analyst needs to describe the positive and negative aspects of an object and complex fuzzy set (CFS) which a decision analyst needs to handle two-dimensional (two variables) information. When there is information of two variables with positive and negative aspects then a decision analyst needs BCFS to handle this information. In this article, we also interpreted some basic operations on BCFS like a complement, intersection, and union and explained them with the help of examples. Additionally, we defined the concept of type-1 partially BCFS and type-2 partially BCFS. Further, we interpreted some generalized trigonometric similarity measures such as generalized cosine similarity measure, generalized tangent similarity measure, generalized cotangent similarity measure, and generalized hybrid trigonometric similarity measure for BCFS. The weighted generalized trigonometric similarity measures are also presented in this article. After that, we applied these similarity measures (SMs) in two real-life applications (pattern recognition and medical diagnosis) to show the benefits and advantages of our proposed SMs. Finally, we did a comparison of our demonstrated SMs with some existing SMs to show the superiority, usefulness, and effectiveness of our proposed SMs.

中文翻译：

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双极复杂模糊集的一种新方法及其在广义相似性度量中的应用

双极复数模糊集 (BCFS) 和复双极模糊集 (CBFS) 的概念已经给出，但是 BCFS 和 CBFS 的这些概念存在的问题是它们与我们在本文中讨论的复数的基本定义相矛盾，然后我们定义了 BCFS 的新定义。与现有的 BCFS 和 CBFS 相比，我们定义的 BCFS 概念更接近于双极性，并且更准确。BCFS是决策分析师需要描述对象的正反面的双极模糊集（BFS）和决策分析师需要处理二维（两个变量）信息的复杂模糊集（CFS）的融合。当存在具有积极和消极方面的两个变量的信息时，决策分析师需要 BCFS 来处理这些信息。在本文中，我们还解释了 BCFS 上的一些基本操作，例如补码、交集和并集，并通过示例进行了解释。此外，我们定义了类型 1 部分 BCFS 和类型 2 部分 BCFS 的概念。此外，我们解释了一些广义三角相似性度量，例如广义余弦相似性度量、广义正切相似性度量、广义余切相似性度量和 BCFS 的广义混合三角相似性度量。本文还介绍了加权广义三角相似性度量。之后，我们在两个实际应用（模式识别和医学诊断）中应用了这些相似性度量（SM），以展示我们提出的 SM 的好处和优势。最后，