{"title":"作为关系的法律:","authors":"","doi":"10.2307/j.ctv1q26ntn.5","DOIUrl":null,"url":null,"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:","PeriodicalId":208281,"journal":{"name":"The Mind of God and the Works of Nature","volume":"1 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"2019-02-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"LAWS AS RELATIONS:\",\"authors\":\"\",\"doi\":\"10.2307/j.ctv1q26ntn.5\",\"DOIUrl\":null,\"url\":null,\"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:\",\"PeriodicalId\":208281,\"journal\":{\"name\":\"The Mind of God and the Works of Nature\",\"volume\":\"1 1\",\"pages\":\"0\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2019-02-07\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"The Mind of God and the Works of Nature\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.2307/j.ctv1q26ntn.5\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"The Mind of God and the Works of Nature","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.2307/j.ctv1q26ntn.5","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
摘要
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:
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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