$$x \in A \Leftrightarrow x \notin \complement_U A$$

$$x \in \complement_U A \Leftrightarrow x \notin A$$

$$\complement_U(A \cap B) = \complement_U A \cup \complement_U B$$

$$\complement_U(A \cup B) = \complement_U A \cap \complement_U B$$

$$A \cap B = A \Leftrightarrow A \cup B = B \Leftrightarrow A \subseteq B \Leftrightarrow \complement_U B \subseteq \complement_U A$$

$$A \cap \complement_U B = \varnothing \Leftrightarrow \complement_U A \cup B = R$$

$$\text{card}(A \cup B) = \text{card}A + \text{card}B - \text{card}(A \cap B)$$

集合 $\{a_1, a_2, \cdots, a_n\}$ 的子集个数有 $2^n$ 个

真子集有 $2^n - 1$ 个；非空子集有 $2^n - 1$ 个；非空真子集有 $2^n - 2$ 个

设 $x_1, x_2 \in [a, b]$，$x_1 \neq x_2$，则

$$(x_1 - x_2)[f(x_1) - f(x_2)] > 0 \Leftrightarrow \frac{f(x_1) - f(x_2)}{x_1 - x_2} > 0 \Leftrightarrow f(x) \text{在}[a, b]\text{上递增}$$

$$(x_1 - x_2)[f(x_1) - f(x_2)] < 0 \Leftrightarrow \frac{f(x_1) - f(x_2)}{x_1 - x_2} < 0 \Leftrightarrow f(x) \text{在}[a, b]\text{上递减}$$

奇函数：$f(-x) = -f(x)$，图象关于原点对称

偶函数：$f(-x) = f(x)$，图象关于y轴对称

奇函数常见形式：$y = ax + \frac{b}{x}, y = x^n$（n为奇数）, y = \sin x, y = \tan x, y = e^x - e^{-x}$

偶函数常见形式：$y = x^n$（n为偶数）, y = \cos x, y = e^x + e^{-x}, y = f(|x|)$

$$f(x + a) = f(x) \Rightarrow T = a$$

$$f(x + a) = -f(x) \Rightarrow T = 2a$$

$$f(x + a) = \frac{1}{f(x)} \Rightarrow T = 2a$$

通项公式：$$a_n = a_1 + (n - 1)d = dn + (a_1 - d)$$

前n项和：$$S_n = \frac{n(a_1 + a_n)}{2} = na_1 + \frac{n(n - 1)}{2}d = \frac{d}{2}n^2 + (a_1 - \frac{d}{2})n$$

通项公式：$$a_n = a_1 q^{n-1} = \frac{a_1}{q} \cdot q^n$$

前n项和：$$S_n = \begin{cases} na_1, & q = 1 \\ \frac{a_1(1 - q^n)}{1 - q}, & q \neq 1 \end{cases}$$

$$1 + 2 + \cdots + n = \frac{n(n + 1)}{2}$$

$$1^2 + 2^2 + \cdots + n^2 = \frac{n(n + 1)(2n + 1)}{6}$$

$$1^3 + 2^3 + \cdots + n^3 = \frac{1}{4}n^2(n + 1)^2$$

$$\sin^2\theta + \cos^2\theta = 1$$

$$\tan\theta = \frac{\sin\theta}{\cos\theta}$$

$$\tan\theta \cdot \cot\theta = 1$$

$$\sin(\pi + \alpha) = -\sin\alpha, \cos(\pi + \alpha) = -\cos\alpha, \tan(\pi + \alpha) = \tan\alpha$$

$$\sin(\pi - \alpha) = \sin\alpha, \cos(\pi - \alpha) = -\cos\alpha, \tan(\pi - \alpha) = -\tan\alpha$$

$$\sin\left(\frac{\pi}{2} - \alpha\right) = \cos\alpha, \cos\left(\frac{\pi}{2} - \alpha\right) = \sin\alpha$$

$$\sin(\alpha \pm \beta) = \sin\alpha\cos\beta \pm \cos\alpha\sin\beta$$

$$\cos(\alpha \pm \beta) = \cos\alpha\cos\beta \mp \sin\alpha\sin\beta$$

$$\tan(\alpha \pm \beta) = \frac{\tan\alpha \pm \tan\beta}{1 \mp \tan\alpha\tan\beta}$$

$$\vec{a} \cdot \vec{b} = |\vec{a}||\vec{b}|\cos\theta = x_1x_2 + y_1y_2$$

$$(\vec{a} \pm \vec{b})^2 = |\vec{a} \pm \vec{b}|^2 = \vec{a}^2 \pm 2\vec{a} \cdot \vec{b} + \vec{b}^2$$

$$|\vec{a} + \vec{b}|^2 + |\vec{a} - \vec{b}|^2 = 2(|\vec{a}|^2 + |\vec{b}|^2)$$

$$\vec{a} \parallel \vec{b} \Leftrightarrow x_1y_2 - x_2y_1 = 0$$

$$\vec{a} \perp \vec{b} \Leftrightarrow \vec{a} \cdot \vec{b} = 0 \Leftrightarrow x_1x_2 + y_1y_2 = 0$$

$$a, b \in R^+ \Rightarrow \frac{a + b}{2} \geq \sqrt{ab}$$

$$\frac{2}{\frac{1}{a} + \frac{1}{b}} \leq \sqrt{ab} \leq \frac{a + b}{2} \leq \sqrt{\frac{a^2 + b^2}{2}}$$

应用条件：一正二定三相等

$$(a^2 + b^2)(c^2 + d^2) \geq (ac + bd)^2$$

$$(a_1^2 + a_2^2 + \cdots + a_n^2)(b_1^2 + \cdots + b_n^2) \geq (a_1b_1 + \cdots + a_nb_n)^2$$

$$k = \tan\alpha = \frac{y_2 - y_1}{x_2 - x_1} \quad (x_1 \neq x_2)$$

倾斜角$\alpha = 0^\circ$时$k=0$；$\alpha = 90^\circ$时斜率不存在

$$d = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}} \quad (\text{点}P(x_0, y_0), \text{直线}Ax + By + C = 0)$$

标准方程：$$(x - a)^2 + (y - b)^2 = r^2$$

一般方程：$$x^2 + y^2 + Dx + Ey + F = 0 \quad (D^2 + E^2 - 4F > 0)$$

参数方程：$$\begin{cases} x = a + r\cos\theta \\ y = b + r\sin\theta \end{cases} \quad (\theta\text{为参数})$$

标准方程：$$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \quad (a > b > 0)$$

焦半径：$$|PF_1| = a + ex, |PF_2| = a - ex$$

离心率：$$e = \frac{c}{a} < 1$$，其中$c^2 = a^2 - b^2$

标准方程：$$\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \quad (a > 0, b > 0)$$

渐近线：$$y = \pm \frac{b}{a}x$$

离心率：$$e = \frac{c}{a} > 1$$，其中$c^2 = a^2 + b^2$

标准方程：$$y^2 = 2px \quad (p > 0)$$

焦半径：$$|CF| = x_0 + \frac{p}{2}$$

通径长：$$2p$$

设$\vec{a} = (x_1, y_1, z_1), \vec{b} = (x_2, y_2, z_2)$

$$\vec{a} \cdot \vec{b} = x_1x_2 + y_1y_2 + z_1z_2$$

$$\cos\langle\vec{a}, \vec{b}\rangle = \frac{x_1x_2 + y_1y_2 + z_1z_2}{\sqrt{x_1^2 + y_1^2 + z_1^2}\sqrt{x_2^2 + y_2^2 + z_2^2}}$$

$$d = \frac{|\vec{AP} \cdot \vec{n}|}{|\vec{n}|} \quad (\vec{n}\text{为平面法向量}, A\text{为平面内一点}, P\text{为平面外一点})$$

柱体：$$V = Sh$$

锥体：$$V = \frac{1}{3}Sh$$

球体：$$V = \frac{4}{3}\pi R^3, S = 4\pi R^2$$

排列数：$$A_n^m = \frac{n!}{(n - m)!} = n(n - 1)\cdots(n - m + 1)$$

组合数：$$C_n^m = \frac{n!}{m!(n - m)!} = \frac{A_n^m}{A_m^m}$$

组合数性质：$$C_n^m = C_n^{n - m}, C_n^m + C_n^{m - 1} = C_{n + 1}^m$$

$$(a + b)^n = \sum_{k=0}^n C_n^k a^{n - k}b^k$$

通项公式：$$T_{r + 1} = C_n^r a^{n - r}b^r$$

$$P(A) = \frac{k}{n}$$

其中$n$为样本空间样本点总数，$k$为事件$A$包含的样本点数

$$P(AB) = P(A)P(B)$$

$$P_n(k) = C_n^k p^k(1 - p)^{n - k}$$

期望：$$E(\xi) = np$$

方差：$$D(\xi) = np(1 - p)$$

$$(c)' = 0, (x^\alpha)' = \alpha x^{\alpha - 1}$$

$$(\sin x)' = \cos x, (\cos x)' = -\sin x$$

$$(a^x)' = a^x\ln a, (e^x)' = e^x$$

$$(\log_a x)' = \frac{1}{x\ln a}, (\ln x)' = \frac{1}{x}$$

$$(u \pm v)' = u' \pm v'$$

$$(uv)' = u'v + uv'$$

$$\left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2} \quad (v \neq 0)$$

$$f'(x) > 0 \Rightarrow f(x)\text{单调递增}$$

$$f'(x) < 0 \Rightarrow f(x)\text{单调递减}$$

设$z_1 = a + bi, z_2 = c + di$

$$z_1 + z_2 = (a + c) + (b + d)i$$

$$z_1 - z_2 = (a - c) + (b - d)i$$

$$z_1z_2 = (ac - bd) + (bc + ad)i$$

$$\frac{z_1}{z_2} = \frac{ac + bd}{c^2 + d^2} + \frac{bc - ad}{c^2 + d^2}i \quad (z_2 \neq 0)$$

$$|z| = |a + bi| = \sqrt{a^2 + b^2}$$

均值不等式链：$$\frac{2}{\frac{1}{a} + \frac{1}{b}} \leq \sqrt{ab} \leq \frac{a + b}{2} \leq \sqrt{\frac{a^2 + b^2}{2}}$$

正弦定理：$$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R$$

余弦定理：$$a^2 = b^2 + c^2 - 2bc\cos A$$

弦长公式：$$|AB| = \sqrt{1 + k^2}|x_1 - x_2| = \sqrt{(1 + k^2)[(x_1 + x_2)^2 - 4x_1x_2]}$$

指数放缩：$$e^x \geq 1 + x, e^x \geq ex$$

对数放缩：$$\ln x \leq x - 1, \ln x \geq 1 - \frac{1}{x}$$

三角放缩：$$\sin x \leq x \leq \tan x \quad (0 \leq x < \frac{\pi}{2})$$