一、选择题(每题5分,共40分)在每小题给出的四个选项中,只有一项是符合题目要求的.
-
-
2.
对于任意实数
a ,
b ,
c ,
d , 有以下四个命题:
①若
, 则
; ②若
, 则
;
③若
, 则
; ④若
, 则![](//math.21cnjy.com/MathMLToImage?mml=%3Cmath+xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmfrac%3E%3Cmrow%3E%3Cmn%3E1%3C%2Fmn%3E%3C%2Fmrow%3E%3Cmrow%3E%3Cmi%3Ea%3C%2Fmi%3E%3C%2Fmrow%3E%3C%2Fmfrac%3E%3Cmo%3E%26gt%3B%3C%2Fmo%3E%3Cmfrac%3E%3Cmrow%3E%3Cmn%3E1%3C%2Fmn%3E%3C%2Fmrow%3E%3Cmrow%3E%3Cmi%3Eb%3C%2Fmi%3E%3C%2Fmrow%3E%3C%2Fmfrac%3E%3C%2Fmath%3E)
其中正确的有( )
A . 1个
B . 2个
C . 3个
D . 4个
-
3.
“方程
![](//math.21cnjy.com/MathMLToImage?mml=%3Cmath+xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmsup%3E%3Cmrow%3E%3Cmi%3Ex%3C%2Fmi%3E%3C%2Fmrow%3E%3Cmrow%3E%3Cmn%3E2%3C%2Fmn%3E%3C%2Fmrow%3E%3C%2Fmsup%3E%3Cmo%3E%E2%88%92%3C%2Fmo%3E%3Cmn%3E2%3C%2Fmn%3E%3Cmi%3Ex%3C%2Fmi%3E%3Cmo%3E%2B%3C%2Fmo%3E%3Cmi%3Em%3C%2Fmi%3E%3Cmo%3E%3D%3C%2Fmo%3E%3Cmn%3E0%3C%2Fmn%3E%3C%2Fmath%3E)
至多有一个实数解”的一个充要条件是( )
-
4.
已知条件
p:
![](//math.21cnjy.com/MathMLToImage?mml=%3Cmath+xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmo%3E%E2%88%92%3C%2Fmo%3E%3Cmn%3E1%3C%2Fmn%3E%3Cmo%3E%26lt%3B%3C%2Fmo%3E%3Cmi%3Ex%3C%2Fmi%3E%3Cmo%3E%26lt%3B%3C%2Fmo%3E%3Cmn%3E1%3C%2Fmn%3E%3C%2Fmath%3E)
,
q:
![](//math.21cnjy.com/MathMLToImage?mml=%3Cmath+xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmi%3Ex%3C%2Fmi%3E%3Cmo%3E%26gt%3B%3C%2Fmo%3E%3Cmi%3Em%3C%2Fmi%3E%3C%2Fmath%3E)
, 若
p是
q的充分不必要条件,则实数
m的取值范围是( )
-
-
6.
若不等式
![](//math.21cnjy.com/MathMLToImage?mml=%3Cmath+xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmn%3E2%3C%2Fmn%3E%3Cmi%3Ek%3C%2Fmi%3E%3Cmsup%3E%3Cmrow%3E%3Cmi%3Ex%3C%2Fmi%3E%3C%2Fmrow%3E%3Cmrow%3E%3Cmn%3E2%3C%2Fmn%3E%3C%2Fmrow%3E%3C%2Fmsup%3E%3Cmo%3E%2B%3C%2Fmo%3E%3Cmi%3Ek%3C%2Fmi%3E%3Cmi%3Ex%3C%2Fmi%3E%3Cmo%3E%E2%88%92%3C%2Fmo%3E%3Cmfrac%3E%3Cmrow%3E%3Cmn%3E3%3C%2Fmn%3E%3C%2Fmrow%3E%3Cmrow%3E%3Cmn%3E8%3C%2Fmn%3E%3C%2Fmrow%3E%3C%2Fmfrac%3E%3Cmo%3E%26lt%3B%3C%2Fmo%3E%3Cmn%3E0%3C%2Fmn%3E%3C%2Fmath%3E)
对一切实数
x都成立,则
k的取值范围为( )
-
7.
某城市数、理、化竞赛时,高一某班有26名学生参加数学竞赛,25名学生参加物理竞赛,23名学生参加化学竞赛,其中参加数、理、化三科竞赛的有7名,只参加数、物两科的有6名,只参加物、化两科的有8名,只参加数、化两科的有5名.若该班学生共有51名,则没有参加任何竞赛的学生共有( )名
A . 7
B . 8
C . 9
D . 10
-
8.
若不等式组
![](//math.21cnjy.com/MathMLToImage?mml=%3Cmath+xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmrow%3E%3Cmo%3E%7B%3C%2Fmo%3E%3Cmtable+columnalign%3D%22left%22%3E%3Cmtr%3E%3Cmtd%3E%3Cmrow%3E%3Cmsup%3E%3Cmrow%3E%3Cmi%3Ex%3C%2Fmi%3E%3C%2Fmrow%3E%3Cmrow%3E%3Cmn%3E2%3C%2Fmn%3E%3C%2Fmrow%3E%3C%2Fmsup%3E%3Cmo%3E%E2%88%92%3C%2Fmo%3E%3Cmn%3E2%3C%2Fmn%3E%3Cmi%3Ex%3C%2Fmi%3E%3Cmo%3E%E2%88%92%3C%2Fmo%3E%3Cmn%3E3%3C%2Fmn%3E%3Cmo%3E%E2%89%A4%3C%2Fmo%3E%3Cmn%3E0%3C%2Fmn%3E%3C%2Fmrow%3E%3C%2Fmtd%3E%3C%2Fmtr%3E%3Cmtr%3E%3Cmtd%3E%3Cmrow%3E%3Cmsup%3E%3Cmrow%3E%3Cmi%3Ex%3C%2Fmi%3E%3C%2Fmrow%3E%3Cmrow%3E%3Cmn%3E2%3C%2Fmn%3E%3C%2Fmrow%3E%3C%2Fmsup%3E%3Cmo%3E%2B%3C%2Fmo%3E%3Cmn%3E4%3C%2Fmn%3E%3Cmi%3Ex%3C%2Fmi%3E%3Cmo%3E%E2%88%92%3C%2Fmo%3E%3Cmrow%3E%3Cmo%3E%28%3C%2Fmo%3E%3Cmrow%3E%3Cmn%3E1%3C%2Fmn%3E%3Cmo%3E%2B%3C%2Fmo%3E%3Cmi%3Ea%3C%2Fmi%3E%3C%2Fmrow%3E%3Cmo%3E%29%3C%2Fmo%3E%3C%2Fmrow%3E%3Cmo%3E%E2%89%A4%3C%2Fmo%3E%3Cmn%3E0%3C%2Fmn%3E%3C%2Fmrow%3E%3C%2Fmtd%3E%3C%2Fmtr%3E%3C%2Fmtable%3E%3C%2Fmrow%3E%3C%2Fmath%3E)
的解集不是空集,则实数
a的取值范围是( )
二、选择题:本题共4小题,每小题5分,共20分.在每小题给出的选项中,有多项符合题目要求.全部选对的得5分,部分选对的得2分,有选错的得0分.
三、填空题:本题共4小题,每小题5分,共20分.
-
13.
命题:
![](//math.21cnjy.com/MathMLToImage?mml=%3Cmath+xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmo%3E%E2%88%83%3C%2Fmo%3E%3Cmi%3Ex%3C%2Fmi%3E%3Cmo%3E%E2%88%88%3C%2Fmo%3E%3Cmi%3EQ%3C%2Fmi%3E%3C%2Fmath%3E)
,
![](//math.21cnjy.com/MathMLToImage?mml=%3Cmath+xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmsup%3E%3Cmrow%3E%3Cmi%3Ex%3C%2Fmi%3E%3C%2Fmrow%3E%3Cmrow%3E%3Cmn%3E3%3C%2Fmn%3E%3C%2Fmrow%3E%3C%2Fmsup%3E%3Cmo%3E%2B%3C%2Fmo%3E%3Cmi%3Ex%3C%2Fmi%3E%3Cmo%3E%E2%89%A0%3C%2Fmo%3E%3Cmn%3E0%3C%2Fmn%3E%3C%2Fmath%3E)
的否定是
.
-
-
15.
已知集合
![](//math.21cnjy.com/MathMLToImage?mml=%3Cmath+xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmi%3EA%3C%2Fmi%3E%3Cmo%3E%3D%3C%2Fmo%3E%3Cmrow%3E%3Cmo%3E%7B%3C%2Fmo%3E%3Cmrow%3E%3Cmi%3Ex%3C%2Fmi%3E%3Cmrow%3E%3Cmo%3E%7C%3C%2Fmo%3E%3Cmrow%3E%3Cmi%3Em%3C%2Fmi%3E%3Cmsup%3E%3Cmrow%3E%3Cmi%3Ex%3C%2Fmi%3E%3C%2Fmrow%3E%3Cmrow%3E%3Cmn%3E2%3C%2Fmn%3E%3C%2Fmrow%3E%3C%2Fmsup%3E%3Cmo%3E%2B%3C%2Fmo%3E%3Cmn%3E2%3C%2Fmn%3E%3Cmi%3Ex%3C%2Fmi%3E%3Cmo%3E%2B%3C%2Fmo%3E%3Cmn%3E2%3C%2Fmn%3E%3Cmo%3E%3D%3C%2Fmo%3E%3Cmn%3E0%3C%2Fmn%3E%3C%2Fmrow%3E%3C%2Fmrow%3E%3C%2Fmrow%3E%3Cmo%3E%7D%3C%2Fmo%3E%3C%2Fmrow%3E%3C%2Fmath%3E)
中有两个元素,则实数
m满足的条件为
.
-
16.
已知
![](//math.21cnjy.com/MathMLToImage?mml=%3Cmath+xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmi%3EA%3C%2Fmi%3E%3Cmo%3E%3D%3C%2Fmo%3E%3Cmn%3E1%3C%2Fmn%3E%3Cmo%3E%2B%3C%2Fmo%3E%3Cmsup%3E%3Cmrow%3E%3Cmi%3Ea%3C%2Fmi%3E%3C%2Fmrow%3E%3Cmrow%3E%3Cmn%3E2%3C%2Fmn%3E%3C%2Fmrow%3E%3C%2Fmsup%3E%3C%2Fmath%3E)
,
![](//math.21cnjy.com/MathMLToImage?mml=%3Cmath+xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmi%3EB%3C%2Fmi%3E%3Cmo%3E%3D%3C%2Fmo%3E%3Cmn%3E1%3C%2Fmn%3E%3Cmo%3E%E2%88%92%3C%2Fmo%3E%3Cmi%3Ea%3C%2Fmi%3E%3C%2Fmath%3E)
,
![](//math.21cnjy.com/MathMLToImage?mml=%3Cmath+xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmi%3EC%3C%2Fmi%3E%3Cmo%3E%3D%3C%2Fmo%3E%3Cmfrac%3E%3Cmrow%3E%3Cmn%3E1%3C%2Fmn%3E%3C%2Fmrow%3E%3Cmrow%3E%3Cmn%3E1%3C%2Fmn%3E%3Cmo%3E%2B%3C%2Fmo%3E%3Cmi%3Ea%3C%2Fmi%3E%3C%2Fmrow%3E%3C%2Fmfrac%3E%3C%2Fmath%3E)
,
![](//math.21cnjy.com/MathMLToImage?mml=%3Cmath+xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmi%3ED%3C%2Fmi%3E%3Cmo%3E%3D%3C%2Fmo%3E%3Cmfrac%3E%3Cmrow%3E%3Cmn%3E1%3C%2Fmn%3E%3C%2Fmrow%3E%3Cmrow%3E%3Cmn%3E1%3C%2Fmn%3E%3Cmo%3E%E2%88%92%3C%2Fmo%3E%3Cmi%3Ea%3C%2Fmi%3E%3C%2Fmrow%3E%3C%2Fmfrac%3E%3C%2Fmath%3E)
为四个互不相等的实数.若
A ,
B ,
C ,
D中
C最大,则实数
a的取值范围为
.
四、解答题:本题共6小题,共70分.解答应写出文字说明、证明过程或演算步骤.
-
-
-
-
(2)
求不等式
![](//math.21cnjy.com/MathMLToImage?mml=%3Cmath+xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmrow%3E%3Cmi%3Ec%3C%2Fmi%3E%3Cmsup%3E%3Cmi%3Ex%3C%2Fmi%3E%3Cmn%3E2%3C%2Fmn%3E%3C%2Fmsup%3E%3Cmo%3E%2B%3C%2Fmo%3E%3Cmi%3Eb%3C%2Fmi%3E%3Cmi%3Ex%3C%2Fmi%3E%3Cmo%3E%2B%3C%2Fmo%3E%3Cmn%3E1%3C%2Fmn%3E%3Cmo%3E%E2%89%A4%3C%2Fmo%3E%3Cmn%3E0%3C%2Fmn%3E%3C%2Fmrow%3E%3C%2Fmath%3E)
的解集.
-
19.
动物园要以墙体为背面,用钢筋网围成四间具有相同面积的矩形虎笼.
![](//tikupic.21cnjy.com/2023/12/29/82/17/8217decd09559b35440500b4fee50230_m_232x58.png)
-
(1)
现有36m长的钢筋网材料;x , y的值分别为多少时,每间虎笼的面积最大,最大值为多少?
-
(2)
若每间虎笼的面积为
![](//math.21cnjy.com/MathMLToImage?mml=%3Cmath+xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmn%3E2%3C%2Fmn%3E%3Cmn%3E0%3C%2Fmn%3E%3Cmsup%3E%3Cmrow%3E%3Cmtext%3Em%3C%2Fmtext%3E%3C%2Fmrow%3E%3Cmrow%3E%3Cmn%3E2%3C%2Fmn%3E%3C%2Fmrow%3E%3C%2Fmsup%3E%3C%2Fmath%3E)
,
x ,
y的值分别为多少时,围成四间虎笼的钢筋网总长最小,最小值是多少?
-
20.
已知集合
![](//math.21cnjy.com/MathMLToImage?mml=%3Cmath+xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmi%3EA%3C%2Fmi%3E%3Cmo%3E%3D%3C%2Fmo%3E%3Cmrow%3E%3Cmo%3E%7B%3C%2Fmo%3E%3Cmrow%3E%3Cmn%3E2%3C%2Fmn%3E%3Cmn%3E++%EF%BC%8C+%3C%2Fmn%3E%3Cmn%3E6%3C%2Fmn%3E%3C%2Fmrow%3E%3Cmo%3E%7D%3C%2Fmo%3E%3C%2Fmrow%3E%3C%2Fmath%3E)
.
-
(1)
若集合
![](//math.21cnjy.com/MathMLToImage?mml=%3Cmath+xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmi%3EB%3C%2Fmi%3E%3Cmo%3E%3D%3C%2Fmo%3E%3Cmrow%3E%3Cmo%3E%7B%3C%2Fmo%3E%3Cmrow%3E%3Cmi%3Ea%3C%2Fmi%3E%3Cmo%3E%2B%3C%2Fmo%3E%3Cmn%3E1%3C%2Fmn%3E%3Cmn%3E++%EF%BC%8C+%3C%2Fmn%3E%3Cmsup%3E%3Cmrow%3E%3Cmi%3Ea%3C%2Fmi%3E%3C%2Fmrow%3E%3Cmrow%3E%3Cmn%3E2%3C%2Fmn%3E%3C%2Fmrow%3E%3C%2Fmsup%3E%3Cmo%3E%E2%88%92%3C%2Fmo%3E%3Cmn%3E2%3C%2Fmn%3E%3Cmn%3E3%3C%2Fmn%3E%3C%2Fmrow%3E%3Cmo%3E%7D%3C%2Fmo%3E%3C%2Fmrow%3E%3C%2Fmath%3E)
, 且
![](//math.21cnjy.com/MathMLToImage?mml=%3Cmath+xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmi%3EA%3C%2Fmi%3E%3Cmo%3E%3D%3C%2Fmo%3E%3Cmi%3EB%3C%2Fmi%3E%3C%2Fmath%3E)
, 求
a的值;
-
(2)
若集合
![](//math.21cnjy.com/MathMLToImage?mml=%3Cmath+xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmi%3EC%3C%2Fmi%3E%3Cmo%3E%3D%3C%2Fmo%3E%3Cmrow%3E%3Cmo%3E%7B%3C%2Fmo%3E%3Cmrow%3E%3Cmi%3Ex%3C%2Fmi%3E%3Cmrow%3E%3Cmo%3E%7C%3C%2Fmo%3E%3Cmrow%3E%3Cmi%3Ea%3C%2Fmi%3E%3Cmsup%3E%3Cmrow%3E%3Cmi%3Ex%3C%2Fmi%3E%3C%2Fmrow%3E%3Cmrow%3E%3Cmn%3E2%3C%2Fmn%3E%3C%2Fmrow%3E%3C%2Fmsup%3E%3Cmo%3E%E2%88%92%3C%2Fmo%3E%3Cmi%3Ex%3C%2Fmi%3E%3Cmo%3E%2B%3C%2Fmo%3E%3Cmn%3E6%3C%2Fmn%3E%3Cmo%3E%3D%3C%2Fmo%3E%3Cmn%3E0%3C%2Fmn%3E%3C%2Fmrow%3E%3C%2Fmrow%3E%3C%2Fmrow%3E%3Cmo%3E%7D%3C%2Fmo%3E%3C%2Fmrow%3E%3C%2Fmath%3E)
, 且
A与
C有包含关系,求
a的取值范围.
-
21.
设
![](//math.21cnjy.com/MathMLToImage?mml=%3Cmath+xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmi%3Ey%3C%2Fmi%3E%3Cmo%3E%3D%3C%2Fmo%3E%3Cmi%3Ea%3C%2Fmi%3E%3Cmsup%3E%3Cmrow%3E%3Cmi%3Ex%3C%2Fmi%3E%3C%2Fmrow%3E%3Cmrow%3E%3Cmn%3E2%3C%2Fmn%3E%3C%2Fmrow%3E%3C%2Fmsup%3E%3Cmo%3E%2B%3C%2Fmo%3E%3Cmrow%3E%3Cmo%3E%28%3C%2Fmo%3E%3Cmrow%3E%3Cmn%3E1%3C%2Fmn%3E%3Cmo%3E%E2%88%92%3C%2Fmo%3E%3Cmi%3Ea%3C%2Fmi%3E%3C%2Fmrow%3E%3Cmo%3E%29%3C%2Fmo%3E%3C%2Fmrow%3E%3Cmi%3Ex%3C%2Fmi%3E%3Cmo%3E%2B%3C%2Fmo%3E%3Cmi%3Ea%3C%2Fmi%3E%3Cmo%3E%E2%88%92%3C%2Fmo%3E%3Cmn%3E2%3C%2Fmn%3E%3C%2Fmath%3E)
.
-
(1)
命题
p:
![](//math.21cnjy.com/MathMLToImage?mml=%3Cmath+xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmo%3E%E2%88%83%3C%2Fmo%3E%3Cmi%3Ex%3C%2Fmi%3E%3Cmo%3E%E2%88%88%3C%2Fmo%3E%3Cmtext%3ER%3C%2Fmtext%3E%3C%2Fmath%3E)
, 使得
![](//math.21cnjy.com/MathMLToImage?mml=%3Cmath+xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmi%3Ey%3C%2Fmi%3E%3Cmo%3E%26lt%3B%3C%2Fmo%3E%3Cmo%3E%E2%88%92%3C%2Fmo%3E%3Cmn%3E2%3C%2Fmn%3E%3C%2Fmath%3E)
成立.若
p为假命题,求实数
a的取值范围;
-
(2)
解关于
x的不等式
![](//math.21cnjy.com/MathMLToImage?mml=%3Cmath+xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmi%3Ea%3C%2Fmi%3E%3Cmsup%3E%3Cmrow%3E%3Cmi%3Ex%3C%2Fmi%3E%3C%2Fmrow%3E%3Cmrow%3E%3Cmn%3E2%3C%2Fmn%3E%3C%2Fmrow%3E%3C%2Fmsup%3E%3Cmo%3E%2B%3C%2Fmo%3E%3Cmrow%3E%3Cmo%3E%28%3C%2Fmo%3E%3Cmrow%3E%3Cmn%3E1%3C%2Fmn%3E%3Cmo%3E%E2%88%92%3C%2Fmo%3E%3Cmi%3Ea%3C%2Fmi%3E%3C%2Fmrow%3E%3Cmo%3E%29%3C%2Fmo%3E%3C%2Fmrow%3E%3Cmi%3Ex%3C%2Fmi%3E%3Cmo%3E%2B%3C%2Fmo%3E%3Cmi%3Ea%3C%2Fmi%3E%3Cmo%3E%E2%88%92%3C%2Fmo%3E%3Cmn%3E2%3C%2Fmn%3E%3Cmo%3E%26lt%3B%3C%2Fmo%3E%3Cmi%3Ea%3C%2Fmi%3E%3Cmo%3E%E2%88%92%3C%2Fmo%3E%3Cmn%3E1%3C%2Fmn%3E%3Cmrow%3E%3Cmo%3E%28%3C%2Fmo%3E%3Cmrow%3E%3Cmi%3Ea%3C%2Fmi%3E%3Cmo%3E%E2%88%88%3C%2Fmo%3E%3Cmtext%3ER%3C%2Fmtext%3E%3C%2Fmrow%3E%3Cmo%3E%29%3C%2Fmo%3E%3C%2Fmrow%3E%3C%2Fmath%3E)
.
-
22.
已知集合
![](//math.21cnjy.com/MathMLToImage?mml=%3Cmath+xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmi%3EA%3C%2Fmi%3E%3Cmo%3E%3D%3C%2Fmo%3E%3Cmrow%3E%3Cmo%3E%7B%3C%2Fmo%3E%3Cmrow%3E%3Cmsub%3E%3Cmrow%3E%3Cmi%3Ea%3C%2Fmi%3E%3C%2Fmrow%3E%3Cmrow%3E%3Cmn%3E1%3C%2Fmn%3E%3C%2Fmrow%3E%3C%2Fmsub%3E%3Cmn%3E++%EF%BC%8C+%3C%2Fmn%3E%3Cmsub%3E%3Cmrow%3E%3Cmi%3Ea%3C%2Fmi%3E%3C%2Fmrow%3E%3Cmrow%3E%3Cmn%3E2%3C%2Fmn%3E%3C%2Fmrow%3E%3C%2Fmsub%3E%3Cmn%3E++%EF%BC%8C+%3C%2Fmn%3E%3Cmo%3E%E2%8B%85%3C%2Fmo%3E%3Cmo%3E%E2%8B%85%3C%2Fmo%3E%3Cmo%3E%E2%8B%85%3C%2Fmo%3E%3Cmn%3E++%EF%BC%8C+%3C%2Fmn%3E%3Cmsub%3E%3Cmrow%3E%3Cmi%3Ea%3C%2Fmi%3E%3C%2Fmrow%3E%3Cmrow%3E%3Cmi%3En%3C%2Fmi%3E%3C%2Fmrow%3E%3C%2Fmsub%3E%3C%2Fmrow%3E%3Cmo%3E%7D%3C%2Fmo%3E%3C%2Fmrow%3E%3Cmrow%3E%3Cmo%3E%28%3C%2Fmo%3E%3Cmrow%3E%3Cmn%3E0%3C%2Fmn%3E%3Cmo%3E%E2%89%A4%3C%2Fmo%3E%3Cmsub%3E%3Cmrow%3E%3Cmi%3Ea%3C%2Fmi%3E%3C%2Fmrow%3E%3Cmrow%3E%3Cmn%3E1%3C%2Fmn%3E%3C%2Fmrow%3E%3C%2Fmsub%3E%3Cmo%3E%26lt%3B%3C%2Fmo%3E%3Cmsub%3E%3Cmrow%3E%3Cmi%3Ea%3C%2Fmi%3E%3C%2Fmrow%3E%3Cmrow%3E%3Cmn%3E2%3C%2Fmn%3E%3C%2Fmrow%3E%3C%2Fmsub%3E%3Cmo%3E%26lt%3B%3C%2Fmo%3E%3Cmo%3E%E2%8B%85%3C%2Fmo%3E%3Cmo%3E%E2%8B%85%3C%2Fmo%3E%3Cmo%3E%E2%8B%85%3C%2Fmo%3E%3Cmo%3E%26lt%3B%3C%2Fmo%3E%3Cmsub%3E%3Cmrow%3E%3Cmi%3Ea%3C%2Fmi%3E%3C%2Fmrow%3E%3Cmrow%3E%3Cmi%3En%3C%2Fmi%3E%3C%2Fmrow%3E%3C%2Fmsub%3E%3Cmn%3E++%EF%BC%8C+%3C%2Fmn%3E%3Cmi%3En%3C%2Fmi%3E%3Cmo%3E%E2%88%88%3C%2Fmo%3E%3Cmsup%3E%3Cmrow%3E%3Cmtext%3EN%3C%2Fmtext%3E%3C%2Fmrow%3E%3Cmrow%3E%3Cmn%3E%2A%3C%2Fmn%3E%3C%2Fmrow%3E%3C%2Fmsup%3E%3Cmn%3E++%EF%BC%8C+%3C%2Fmn%3E%3Cmi%3En%3C%2Fmi%3E%3Cmo%3E%E2%89%A5%3C%2Fmo%3E%3Cmn%3E3%3C%2Fmn%3E%3C%2Fmrow%3E%3Cmo%3E%29%3C%2Fmo%3E%3C%2Fmrow%3E%3C%2Fmath%3E)
具有性质
P:对任意
![](//math.21cnjy.com/MathMLToImage?mml=%3Cmath+xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmi%3Ei%3C%2Fmi%3E%3Cmn%3E++%EF%BC%8C+%3C%2Fmn%3E%3Cmi%3Ej%3C%2Fmi%3E%3Cmrow%3E%3Cmo%3E%28%3C%2Fmo%3E%3Cmrow%3E%3Cmn%3E1%3C%2Fmn%3E%3Cmo%3E%E2%89%A4%3C%2Fmo%3E%3Cmi%3Ei%3C%2Fmi%3E%3Cmo%3E%E2%89%A4%3C%2Fmo%3E%3Cmi%3Ej%3C%2Fmi%3E%3Cmo%3E%E2%89%A4%3C%2Fmo%3E%3Cmi%3Em%3C%2Fmi%3E%3C%2Fmrow%3E%3Cmo%3E%29%3C%2Fmo%3E%3C%2Fmrow%3E%3C%2Fmath%3E)
,
![](//math.21cnjy.com/MathMLToImage?mml=%3Cmath+xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmsub%3E%3Cmrow%3E%3Cmi%3Ea%3C%2Fmi%3E%3C%2Fmrow%3E%3Cmrow%3E%3Cmi%3Ei%3C%2Fmi%3E%3C%2Fmrow%3E%3C%2Fmsub%3E%3Cmo%3E%2B%3C%2Fmo%3E%3Cmsub%3E%3Cmrow%3E%3Cmi%3Ea%3C%2Fmi%3E%3C%2Fmrow%3E%3Cmrow%3E%3Cmi%3Ej%3C%2Fmi%3E%3C%2Fmrow%3E%3C%2Fmsub%3E%3C%2Fmath%3E)
与
![](//math.21cnjy.com/MathMLToImage?mml=%3Cmath+xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmsub%3E%3Cmrow%3E%3Cmi%3Ea%3C%2Fmi%3E%3C%2Fmrow%3E%3Cmrow%3E%3Cmi%3Ej%3C%2Fmi%3E%3C%2Fmrow%3E%3C%2Fmsub%3E%3Cmo%3E%E2%88%92%3C%2Fmo%3E%3Cmsub%3E%3Cmrow%3E%3Cmi%3Ea%3C%2Fmi%3E%3C%2Fmrow%3E%3Cmrow%3E%3Cmi%3Ei%3C%2Fmi%3E%3C%2Fmrow%3E%3C%2Fmsub%3E%3C%2Fmath%3E)
至少一个属于
A .
-
(1)
分别判断集合
![](//math.21cnjy.com/MathMLToImage?mml=%3Cmath+xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmi%3EM%3C%2Fmi%3E%3Cmo%3E%3D%3C%2Fmo%3E%3Cmrow%3E%3Cmo%3E%7B%3C%2Fmo%3E%3Cmrow%3E%3Cmn%3E0%3C%2Fmn%3E%3Cmn%3E++%EF%BC%8C+%3C%2Fmn%3E%3Cmn%3E2%3C%2Fmn%3E%3Cmn%3E++%EF%BC%8C+%3C%2Fmn%3E%3Cmn%3E4%3C%2Fmn%3E%3C%2Fmrow%3E%3Cmo%3E%7D%3C%2Fmo%3E%3C%2Fmrow%3E%3C%2Fmath%3E)
, 与
![](//math.21cnjy.com/MathMLToImage?mml=%3Cmath+xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmi%3EN%3C%2Fmi%3E%3Cmo%3E%3D%3C%2Fmo%3E%3Cmrow%3E%3Cmo%3E%7B%3C%2Fmo%3E%3Cmrow%3E%3Cmn%3E1%3C%2Fmn%3E%3Cmn%3E++%EF%BC%8C+%3C%2Fmn%3E%3Cmn%3E2%3C%2Fmn%3E%3Cmn%3E++%EF%BC%8C+%3C%2Fmn%3E%3Cmn%3E3%3C%2Fmn%3E%3C%2Fmrow%3E%3Cmo%3E%7D%3C%2Fmo%3E%3C%2Fmrow%3E%3C%2Fmath%3E)
是否具有性质
P , 并说明理由;
-
(2)
证明:
![](//math.21cnjy.com/MathMLToImage?mml=%3Cmath+xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmn%3E0%3C%2Fmn%3E%3Cmo%3E%E2%88%88%3C%2Fmo%3E%3Cmi%3EA%3C%2Fmi%3E%3C%2Fmath%3E)
;
-
(3)
![](//math.21cnjy.com/MathMLToImage?mml=%3Cmath+xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmi%3EA%3C%2Fmi%3E%3Cmo%3E%3D%3C%2Fmo%3E%3Cmrow%3E%3Cmo%3E%7B%3C%2Fmo%3E%3Cmrow%3E%3Cmsub%3E%3Cmrow%3E%3Cmi%3Ea%3C%2Fmi%3E%3C%2Fmrow%3E%3Cmrow%3E%3Cmn%3E1%3C%2Fmn%3E%3C%2Fmrow%3E%3C%2Fmsub%3E%3Cmn%3E++%EF%BC%8C+%3C%2Fmn%3E%3Cmsub%3E%3Cmrow%3E%3Cmi%3Ea%3C%2Fmi%3E%3C%2Fmrow%3E%3Cmrow%3E%3Cmn%3E2%3C%2Fmn%3E%3C%2Fmrow%3E%3C%2Fmsub%3E%3Cmn%3E++%EF%BC%8C+%3C%2Fmn%3E%3Cmsub%3E%3Cmrow%3E%3Cmi%3Ea%3C%2Fmi%3E%3C%2Fmrow%3E%3Cmrow%3E%3Cmn%3E3%3C%2Fmn%3E%3C%2Fmrow%3E%3C%2Fmsub%3E%3C%2Fmrow%3E%3Cmo%3E%7D%3C%2Fmo%3E%3C%2Fmrow%3E%3C%2Fmath%3E)
具有性质
P , 当
![](//math.21cnjy.com/MathMLToImage?mml=%3Cmath+xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmsub%3E%3Cmrow%3E%3Cmi%3Ea%3C%2Fmi%3E%3C%2Fmrow%3E%3Cmrow%3E%3Cmn%3E2%3C%2Fmn%3E%3C%2Fmrow%3E%3C%2Fmsub%3E%3Cmo%3E%3D%3C%2Fmo%3E%3Cmn%3E4%3C%2Fmn%3E%3C%2Fmath%3E)
时,求集合
A .