质数证明!急Consider the following statement:“Prime numbers are integers greater than 1 that are not equal to the product of smaller positive integers.”.Define two sets and interpret the statement as the equally between the two sets.Then prov
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质数证明!急Consider the following statement:“Prime numbers are integers greater than 1 that are not equal to the product of smaller positive integers.”.Define two sets and interpret the statement as the equally between the two sets.Then prov
质数证明!急
Consider the following statement:“Prime numbers are integers greater than 1 that are not equal to the product of smaller positive integers.”.Define two sets and interpret the statement as the equally between the two sets.Then prove that the statement is true by proving that the two sets are equal using double-inclusion.(here,your definition of a prime number is that it is an integer greater than 1 with EXACTLY two posi- tive divisors:1 and itself).译：有一论题：素数是一个大于1且不能写成比它自己小的两个整数之积的整数.根据这个论述,定义两个完全相等的集合.然后通过证明两个集合互相包含,来证明这个结论是正确的.（素数的定义为：素数是一个大于1的整数,且只有两个正除数：1和它本身.
质数证明!急Consider the following statement:“Prime numbers are integers greater than 1 that are not equal to the product of smaller positive integers.”.Define two sets and interpret the statement as the equally between the two sets.Then prov
设集合A={x|x＞1,且x不能写成比它自己小的两个整数之积的整数}
集合B={x|x是一个大于1的整数,且只有两个正除数：1和它本身},即B为质数集合.
如果有一个大于1的正整数x0∈A,那么x0除了1和自己外,没有其他的因数（正除数）.
因为假设x0有除了1和x0自己以为,还有其他的因数a,那么a小于x0,且x0/a是正整数,且x0/a小于0.那么x0=a*（x0/a）可以写成两个小于x0的正整数相乘.与x0∈A矛盾.所以x0除了1和自己外,没有其他的因数.所以x0∈B.所以A包含于B,是B的子集.
如果有1个大于1的正整数x1∈B,即x1只有两个正除数：1和它本身.那么x1不可能成比它自己小的两个整数之积的整数.
因为假设x1可以写成比它自己小的两个整数之积的整数,即x1=ab（a、b是小于x1的正整数）.那么a、b是除了1和x1以为的x1的正除数.这和x1∈B矛盾.所以x1不可能成比它自己小的两个整数之积的整数.所以x1∈A.所以B包含于A,是A的子集.
因为（A包含于B）和（B包含于A）同时成立.所以A=B.所以A集合中的元素都是质数集合的元素,且包括了所有的质数.所以“素数是一个大于1且不能写成比它自己小的两个整数之积的整数.” 的论述是正确的.