设全集S={x∈N+|x≤8},若A∩(CsB)={2,8},(CsA)U(CsB)={1,2,3,4,5,6,7,8},求集合A
设全集S={x∈N+|x≤8},若A∩(CsB)={2,8},(CsA)U(CsB)={1,2,3,4,5,6,7,8},求集合A
数学人气:299 ℃时间:2019-10-10 04:20:40
优质解答
∵S={x∈N+|x≤8}={1,2,3,4,5,6,7,8}
∴(CsA)∪(CsB)={1,2,3,4,5,6,7,8}=S
又∵(CsA)∪(CsB)=Cs(A∩B)//反演律
∴Cs(A∩B)=S
即A∩B=空集//找不到空集符号
又∵A∩(CsB)={2,8}、B∪CsB=S、B∩CsB=空集
∴集合A和集合CsB必含有元素2和8,但集合B必不含有元素2和8
假设集合A除元素2和8之外还含有元素x∈S,则:
集合B必不含有x,//由A∩B=空集知
集合CsB必含有x,因此有://由B∪CsB=S知
A∩(CsB)={2,8,x}
上述结论与题设矛盾,故A只含有元素2和8,由此可知A={2,8}
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