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反对称关系

维基百科,自由的百科全书

数学上,若对所有的 ab 属于 X,下述语句保持有效,则集合 X 上的二元关系 R反对称的:“若 a 关系到 bb 关系到 a,则 a = b。”

数学上表示为:

严格不等是反对称的;实际上 a < bb < a 是不可能的,因此严格不等的反对称性是一种空虚的真英语Vacuous truth

注意,反对称关系不是对称关系aRb 得到 bRa)的反义。有些关系既是对称的又是反对称的,比如"等于"(证明:a=b推出b=a;a=b且b=a推出a=b);有些关系既不是对称的也不是反对称的,比如"爱上……"(证明:a爱b不能推出b爱a;a爱b且b爱a不能推出a和b是同一个人);有些关系是对称的但不是反对称的,比如"和…结婚"(证明:a和b结婚推出b和a结婚;a和b结婚且b和a结婚不能推出a和b是同一个人);有些关系不是对称的但是反对称的,比如正整数的"整除"(证明:3整除6不能推出6整除3;a整除b,即b=ma,m为正整数,且b整除a,即a=nb,n为正整数,则b=ma=mnb,则mn=1且m,n为正整数,则m=n=1,即a=b)。

满足传递性和自反性的反对称关系称为偏序关系

参见

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反对称关系
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