已知集合 A= x -5<x 3 <5 ， B= -3,-1,0,2,3 ，则 A B= （） -1,0 2,3 -3,-1,0 -1,0,2 A 集合的运算集合交集立方根单调性新课标I卷 2024

A 先把 A 化成区间。由 − 5 < x 3 < 5 且 y = x 3 在 R 上单调递增，两边同时开立方得 − 3 5 <path d="M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z"/> < x < 3 5 <path d="M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z"/> 。估计端点大小： 1 3 = 1 < 5 ， 2 3 = 8 > 5 ，所以 1 < 3 5 <path d="M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z"/> < 2 ，即 A ≈ ( − 1.71 , 1.71 ) 。逐个检验 B 中元素是否落入该区间： − 3 不在， − 1 在， 0 在， 2 不在（因为 2 > 3 5 <path d="M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z"/> ）， 3 不在。故 A ∩ B = { − 1 , 0 } ，选 A。

2024 新课标I卷｜集合与常用逻辑用语·集合的运算

已知集合

A = {x ∣ - 5 < x^{3} < 5}

，

B = {- 3, - 1, 0, 2, 3}

，则

A \cap B =

（　　）

{- 1, 0}

{2, 3}

- 3

{- 1, 0, 2}

答案A

分步解析

1先把 $A$ 化成区间。由 $- 5 < x^{3} < 5$ 且 $y = x^{3}$ 在 $R$ 上单调递增，两边同时开立方得 $- 35 < x < 35$ 。
2估计端点大小： $1^{3} = 1 < 5$ ， $2^{3} = 8 > 5$ ，所以 $1 < 35 < 2$ ，即 $A \approx (- 1.71, 1.71)$ 。
3逐个检验 $B$ 中元素是否落入该区间： $- 3$ 不在， $- 1$ 在， $0$ 在， $2$ 不在（因为 $2 > 35$ ）， $3$ 不在。
4故 $A \cap B = {- 1, 0}$ ，选 A。

点评不要急着对

- 5 < x^{3} < 5

两边开方，要说明

y = x^{3}

单调递增才能保号；判断

2

是否入选只需比较

2^{3} = 8

与

5

，不必求出

35

的近似值。

#集合#交集#立方根#单调性

2024 年真题新课标I卷

2024 年新课标I卷·第1题

分步解析

同考点推荐集合与常用逻辑用语 · 集合的运算