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nibaba:高一数学-高一数学集合与简易逻辑复习 精品

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第一章复习与小结(3 课时) 一、知识结构: 本章知识主要分为集合、简单不等式的解法(集合化简) 、简易逻辑三部分: 二、知识回顾: (一)集合 1. 基本概念:集合、元素;有限集、无限集;空集、全集;符号的使用. 2. 集合的表示法:列举法、描述法、图形表示法. 3. 集合元素的特征:确定性、互异性、无序性. 4. 集合运算:交、并、补. 交:A B ? {x | x ? A, 且x ? B} 并:A B ? {x | x ? A或x ? B} 补:C U A ? {x ? U , 且x ? A} 5. 主要性质和运算律 (1) 包含关系: A ? A, ? ? A, A ? U , C U A ?U, A ? B, B ? C ? A ? C; A B ? A, A B ? B; A B ? A, A B ? B. (2) 等价关系: A ? B ? A B ? A ? A B ? B ? C B ?U UA (3) 集合的运算律: 交换律: A ? B ? B ? A; A ? B ? B ? A. 结合律: ( A ? B) ? C ? A ? ( B ? C ); ( A ? B) ? C ? A ? ( B ? C ) 分配律:. A ? ( B ? C ) ? ( A ? B) ? ( A ? C ); A ? ( B ? C ) ? ( A ? B) ? ( A ? C ) 0-1 律: ? A ? ?, ? A ? A,U A ? A,U A ?U 等幂律: A ? A ? A, A ? A ? A. 求补律:A∩?UA=φ A∪?UA=U ?UU=φ ?Uφ =U ?UU(?UA)=A 反演律:?U(A∩B)= (?UA)∪(?UB) ?U(A∪B)= (?UA)∩(?UB) 6. 有限集的元素个数 定义: 有限集 A 的元素的个数叫做集合 A 的基数, 记为 card( A)规定 card(φ ) =0. 基本公式: (1)card ( A B) ? card ( A) ? card ( B) ? card ( A B) (2)card ( A B C ) ? card ( A) ? card ( B) ? card (C ) ? card ( A B) ? card ( B C ) ? card (C ? card ( A B C ) (3) card(?UA)= card(U)- card(A) (二)含绝对值不等式、一元二次不等式的解法及延伸 1.整式不等式的解法 根轴法(零点分段法) ①将不等式化为 a0(x-x1)(x-x2)…(x-xm)>0(<0)形式, 并将各因式 x 的系数化 “+” ;(为了统一方便) ②求根,并在数轴上表示出来; ③由右上方穿线,经过数轴上表示各根的点(为什么?) ; ④若不等式(x 的系数化“+”后)是“>0”,则找“线”在 x 轴上方的区间; 若不等式是“<0”,则找“线”在 x 轴下方的区间. A) x1 x2 x3 x m-3 - x m-2 x m-1 + - xm + x (自右向左正负相间) 则不等式 a0 x n ? a1 x n?1 ? a2 x n?2 ? ?? an ? 0(? 0)(a0 ? 0) 的解可以根据各区间的 符号确定. 特例① 一元一次不等式 ax>b 解的讨论; ②一元二次不等式 ax2+box>0(a>0)解的讨论. ??0 ??0 ??0 二次函数 y ? ax2 ? bx ? c ( a ? 0 )的图象 一元二次方程 有两相异实根 有两相等实根 b x1 ? x2 ? ? 2a ?a ? 0?的根 ax ? bx ? c ? 0 2 x1 , x2 ( x1 ? x2 ) 无实根 ax 2 ? bx ? c ? 0 ( a ? 0)的解集 ax 2 ? bx ? c ? 0 ( a ? 0)的解集 ?x x ? x 或x ? x ? 1 2 ? b? ?x x ? ? ? 2a ? ? ? ? R ?x x 1 ? x ?x 2 ? 2.分式不等式的解法 (1)标准化:移项通分化为 的形式, ( 2 ) f ( x) f ( x) f ( x) f ( x) >0(或 <0); ≥0(或 ≤0) g ( x) g ( x) g ( x) g ( x) 转 化 为 整 式 不 等 式 ( 组 ) f ( x) f ( x) f ( x) g ( x) ? 0 ? 0 ? f ( x) g ( x) ? 0; ?0?? ? g ( x) ? 0 ? g ( x) g ( x) 3.含绝对值不等式的解法 (1)公式法: ax ? b ? c ,与 ax ? b ? c(c ? 0) 型的不等式的解法. (2)定义法:用“零点分区间法”分类讨论. (3)几何法:根据绝对值的几何意义用数形结合思想方法解题. 4.一元二次方程根的分布 一元二次方程 ax2+bx+c=0(a≠0) (1)根的“零分布” :根据判别式和韦达定理分析列式解之. (2)根的“非零分布” :作二次函数图象,用数形结合思想分析列式解之. (三)简易逻辑 1、命题的定义:可以判断真假的语句叫做命题。 2、逻辑联结词、简单命题与复合命题: “或” 、 “且” 、 “非”这些词叫做逻辑联结词;不含有逻辑联结词的命题是 简单命题;由简单命题和逻辑联结词“或” 、 “且” 、 “非”构成的命题是 复合命题。 构成复合命题的形式:p 或 q(记作“p∨q” );p 且 q(记作“p∧q” ); 非 p(记作“┑q” ) 。 3、 “或” 、 “且” 、 “非”的真值判断 (1) “非 p”形式复合命题的真假与 F 的真假相反; (2) “p 且 q”形式复合命题当 P 与 q 同为真时为真,其他情况时为假; (3) “p 或 q”形式复合命题当 p 与

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