Symbols and Notation | 符号与记号

📚 Symbols and Notation | 符号与记号

Mathematics relies on a precise system of symbols and notation to express ideas concisely. In IB Mathematics, mastering these symbols is essential for interpreting problems, writing solutions, and communicating reasoning clearly. This article provides a bilingual overview of the most important symbols and notations used across the syllabus, from sets and logic to calculus and statistics.

数学依赖于精确的符号与记号系统来简洁地表达思想。在IB数学中，掌握这些符号对于理解题目、书写解答和清晰地交流推理至关重要。本文以双语概述了课程大纲中最重要的符号与记号，涵盖集合、逻辑、微积分和统计等领域。

1. Set Notation | 集合符号

The symbol ∈ means ‘is an element of’. For example, x ∈ A reads ‘x belongs to set A’.

符号 ∈ 表示’属于’。例如，x ∈ A 读作’x 属于集合 A’。

The symbol ∉ is the negation: x ∉ A means ‘x is not an element of A’.

符号 ∉ 是其否定：x ∉ A 表示’x 不属于 A’。

A ⊆ B denotes that A is a subset of B; every element of A is also in B. If we wish to emphasise that the subset is proper (A ≠ B), we write A ⊂ B.

A ⊆ B 表示 A 是 B 的子集；A 的每个元素都在 B 中。若要强调是真子集（A ≠ B），则写为 A ⊂ B。

The union of two sets is written A ∪ B, containing all elements from A or B. The intersection is A ∩ B, containing only elements common to both.

两个集合的并集写作 A ∪ B，包含来自 A 或 B 的所有元素。交集写作 A ∩ B，只包含两者共有的元素。

The empty set is denoted by ∅ or {}. It contains no elements.

空集用 ∅ 或 {} 表示，它不包含任何元素。

The complement of set A is often written as A’ or A^c, representing all elements not in A within the universal set U (sometimes ξ).

集合 A 的补集常写作 A’ 或 A^c，表示在全集合 U（有时用 ξ 表示）中不属于 A 的所有元素。

The universal set is the set of all elements under consideration, often denoted U or ξ.

全集合是在所讨论范围内所有元素组成的集合，常用 U 或 ξ 表示。

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2. Logic Symbols | 逻辑符号

The symbol ¬ (or ~) denotes negation: ¬p means ‘not p’, the opposite truth value of p.

符号 ¬（或 ~）表示否定：¬p 意为’非 p’，即 p 的真值取反。

Conjunction p ∧ q means ‘p and q’; both must be true for the compound statement to be true.

合取 p ∧ q 意为’p 且 q’；两者均为真时复合命题才为真。

Disjunction p ∨ q means ‘p or q’ (inclusive or); at least one of them is true.

析取 p ∨ q 意为’p 或 q’（包含性或）；至少一个为真就为真。

Implication p ⇒ q reads ‘if p then q’ or ‘p implies q’. It is false only when p is true and q is false.

蕴含 p ⇒ q 读作’如果 p 则 q’或’p 推出 q’。仅当 p 真而 q 假时才为假。

The biconditional p ⇔ q means ‘p if and only if q’; both sides share the same truth value.

双条件 p ⇔ q 意为’p 当且仅当 q’；两边真值相同。

The universal quantifier ∀ reads ‘for all’ or ‘for every’. For example, ∀x ∈ ℝ, x² ≥ 0.

全称量词 ∀ 读作’对所有’或’对任意’。例如，∀x ∈ ℝ, x² ≥ 0。

The existential quantifier ∃ reads ‘there exists’. ∃x ∈ ℤ such that x² = 4 means there is an integer whose square is 4.

存在量词 ∃ 读作’存在’。∃x ∈ ℤ 使得 x² = 4 表示存在一个整数其平方为 4。

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3. Greek Letters in Mathematics | 数学中的希腊字母

Greek letters are widely used as constants, variables, and operators. The small letter α (alpha) often denotes an angle or a significance level in statistics.

希腊字母被广泛用作常数、变量和运算符号。小写字母 α（阿尔法）常表示一个角或统计中的显著性水平。

β (beta) is frequently used for another angle or the probability of a Type II error.

β（贝塔）常用于表示另一个角或第二类错误的概率。

γ (gamma) can represent an angle, the Euler–Mascheroni constant, or a parameter in distributions.

γ（伽马）可表示角、欧拉-马歇罗尼常数或分布中的参数。

Δ (capital Delta) stands for a change or difference: Δy = y₂ − y₁. It also denotes the discriminant of a quadratic.

大写的 Δ（德尔塔）代表变化量或差值：Δy = y₂ − y₁。它也用于表示二次方程的判别式。

θ (theta) almost universally represents an angle, especially in trigonometry and polar coordinates.

θ（西塔）几乎无一例外地表示一个角，特别是在三角学和极坐标中。

π (pi) is the constant ratio of a circle’s circumference to its diameter, approximately 3.14159.

π（派）是圆的周长与直径之比的常数，约等于 3.14159。

Σ (capital sigma) stands for summation; σ (lower-case sigma) denotes standard deviation in statistics.

大写的 Σ（西格玛）表示求和；小写的 σ（西格玛）表示统计中的标准差。

μ (mu) is the symbol for the population mean, and also used for micro (10⁻⁶) in measurement.

μ（缪）是总体均值的符号，也用于测量中的微（10⁻⁶）。

λ (lambda) commonly denotes an eigenvalue, a decay constant, or a parameter in Poisson distribution.

λ（兰姆达）通常表示特征值、衰减常数或泊松分布中的参数。

ω (omega, lowercase) is used for angular frequency; Ω (capital) often denotes ohm or sample space, depending on context.

小写的 ω（奥米伽）用于角频率；大写的 Ω（奥米伽）依上下文可表示欧姆或样本空间。

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4. Function Notation | 函数记号

A function is written as f: A → B, where A is the domain and B is the codomain. The rule is expressed as f(x).

函数写作 f: A → B，其中 A 是定义域，B 是陪域。对应法则表示为 f(x)。

The image of x under f is f(x). For example, if f(x) = x² + 1, then f(3) = 10.

x 在 f 下的像为 f(x)。例如，若 f(x) = x² + 1，则 f(3) = 10。

The composite function of f and g is (g ∘ f)(x) = g(f(x)). First apply f, then g.

f 与 g 的复合函数记为 (g ∘ f)(x) = g(f(x))。先施加 f，再施加 g。

The inverse function of f is denoted f⁻¹, satisfying f⁻¹(f(x)) = x for all x in the domain where the inverse exists.

f 的反函数记为 f⁻¹，在存在反函数的定义域内满足 f⁻¹(f(x)) = x。

Piecewise functions use a curly bracket to define different expressions over intervals: f(x) = { x² for x < 0, 2x for x ≥ 0 }.

分段函数使用花括号在不同区间定义不同表达式：f(x) = { x² 当 x < 0; 2x 当 x ≥ 0 }。

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5. Limits, Derivatives, and Integrals | 极限、导数与积分

The limit of f(x) as x approaches a is written lim_x→a f(x) = L. This describes the value f(x) approaches near a.

当 x 趋近于 a 时 f(x) 的极限记为 lim_x→a f(x) = L，描述的是 f(x) 在 a 附近趋近的值。

The derivative of a function can be expressed as f'(x) (Lagrange notation) or dy/dx (Leibniz notation). Both represent the instantaneous rate of change.

函数的导数可表示为 f'(x)（拉格朗日记号）或 dy/dx（莱布尼茨记号）。两者都表示瞬时变化率。

The second derivative is written f”(x) or d²y/dx², indicating the rate of change of the gradient.

二阶导数写作 f”(x) 或 d²y/dx²，表示斜率的变化率。

The indefinite integral (antiderivative) is ∫ f(x) dx. The definite integral from a to b is ∫_a^b f(x) dx, representing the signed area under the curve.

不定积分（原函数）记为 ∫ f(x) dx。从 a 到 b 的定积分写作 ∫_a^b f(x) dx，代表曲线下的带符号面积。

The fundamental theorem of calculus links differentiation and integration. The notation [F(x)]_a^b = F(b) − F(a) is used for evaluating definite integrals.

微积分基本定理将微分与积分联系起来。记号 [F(x)]_a^b = F(b) − F(a) 用于计算定积分。

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6. Summation and Product Notation | 求和与求积符号

The Greek capital sigma ∑ is used for summation: ∑_i=1ⁿ a_i = a₁ + a₂ + … + a_n. The index i runs from 1 to n.

希腊大写字母 Σ 用于求和：∑_i=1ⁿ a_i = a₁ + a₂ + … + a_n。索引 i 从 1 变化到 n。

The limits of summation can be any integers; for example, ∑_k=0⁵ 2^k = 1+2+4+8+16+32.

求和上下限可以是任意整数；例如 ∑_k=0⁵ 2<

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