LAWS AS RELATIONS:

The Mind of God and the Works of Nature Pub Date : 2019-02-07 DOI:10.2307/j.ctv1q26ntn.5

引用次数: 0

Abstract

The techniques for proving general laws about relations are similar to those for proving laws of classes. As an example, let’s show that the converse R̆ of any transitive relation R is itself transitive: Let R be any transitive relation, and let x, y, and z be any values in the universe of discourse. Suppose that x bears R̆ to y and y bears R̆ to z. Then, by the definition of ‘converse’, y bears R to x and z bears R to y. Hence, since R is transitive, z also bears R to x. Therefore, again by the definition of ‘converse’, x bears R̆ to z. Since x bears R̆ to z whenever x bears R̆ to y and y bears R̆ to z, R̆ is transitive. Thus the converse of any transitive relation is transitive. Some of the laws of relations state logical connections among properties of relations:

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作为关系的法律:

证明一般关系定律的方法与证明类定律的方法类似。作为一个例子，让我们证明任何传递关系R的逆R _本身是传递的:设R是任何传递关系，设x, y, z是论域中的任何值。假设x从R到y, y从R到z。然后，根据“逆”的定义，y从R到x, z从R到y。因此，由于R是可传递的，z也从R到x。因此，再次根据“逆”的定义，x从R到z。因为x从R到z，无论x从R到y, y从R到z, R是可传递的。因此，任何传递关系的逆都是传递的。一些关系定律描述了关系属性之间的逻辑联系:

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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The Mind of God and the Works of Nature

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