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Notation List for Cambridge International Mathematics Qualifications (For use from 2020) converted by GPT-4o to HTML and MathJax
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| <!DOCTYPE html> | |
| <html lang="en"> | |
| <head> | |
| <meta charset="UTF-8"> | |
| <meta name="viewport" content="width=device-width, initial-scale=1.0"> | |
| <title>Mathematical Notation</title> | |
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| <body> | |
| <h1>Notation List for Cambridge International Mathematics Qualifications</h1> | |
| <h2>1. Set notation</h2> | |
| <ul> | |
| <li>\(\in\) is an element of</li> | |
| <li>\(\notin\) is not an element of</li> | |
| <li>\(\{x_1, x_2, \ldots\}\) the set with elements \(x_1, x_2, \ldots\)</li> | |
| <li>\(\{x : \ldots\}\) the set of all \(x\) such that \(\ldots\)</li> | |
| <li>\(n(A)\) the number of elements in set \(A\)</li> | |
| <li>\(\emptyset\) the empty set</li> | |
| <li>\(U\) the universal set</li> | |
| <li>\(A'\) the complement of the set \(A\)</li> | |
| <li>\(\mathbb{N}\) the set of natural numbers, \(\{1, 2, 3, \ldots\}\)</li> | |
| <li>\(\mathbb{Z}\) the set of integers, \(\{0, \pm1, \pm2, \pm3, \ldots\}\)</li> | |
| <li>\(\mathbb{Q}\) the set of rational numbers, \(\left\{ \frac{p}{q} \middle| p \in \mathbb{Z}, q \neq 0 \right\}\)</li> | |
| <li>\(\mathbb{R}\) the set of real numbers</li> | |
| <li>\(\mathbb{C}\) the set of complex numbers</li> | |
| <li>\((x, y)\) the ordered pair \(x, y\)</li> | |
| <li>\(\subseteq\) is a subset of</li> | |
| <li>\(\subset\) is a proper subset of</li> | |
| <li>\(\cup\) union</li> | |
| <li>\(\cap\) intersection</li> | |
| <li>\([a, b]\) the closed interval \(\{x \in \mathbb{R} : a \leq x \leq b\}\)</li> | |
| <li>\([a, b)\) the interval \(\{x \in \mathbb{R} : a \leq x < b\}\)</li> | |
| <li>\((a, b]\) the interval \(\{x \in \mathbb{R} : a < x \leq b\}\)</li> | |
| <li>\((a, b)\) the open interval \(\{x \in \mathbb{R} : a < x < b\}\)</li> | |
| <li>\((S, \circ)\) the group consisting of the set \(S\) with binary operation \(\circ\)</li> | |
| </ul> | |
| <h2>2. Miscellaneous symbols</h2> | |
| <ul> | |
| <li>= is equal to</li> | |
| <li>≠ is not equal to</li> | |
| <li>≡ is identical to or is congruent to</li> | |
| <li>≈ is approximately equal to</li> | |
| <li>~ is distributed as</li> | |
| <li>≅ is isomorphic to</li> | |
| <li>∝ is proportional to</li> | |
| <li>< is less than</li> | |
| <li>≤ is less than or equal to</li> | |
| <li>> is greater than</li> | |
| <li>≥ is greater than or equal to</li> | |
| <li>∞ infinity</li> | |
| <li>⇒ implies</li> | |
| <li>⇐ is implied by</li> | |
| <li>⇔ implies and is implied by (is equivalent to)</li> | |
| </ul> | |
| <h2>3. Operations</h2> | |
| <ul> | |
| <li>\(a + b\) \(a\) plus \(b\)</li> | |
| <li>\(a - b\) \(a\) minus \(b\)</li> | |
| <li>\(a \times b, ab\) \(a\) multiplied by \(b\)</li> | |
| <li>\(\frac{a}{b}, \frac{a}{b}\) \(a\) divided by \(b\)</li> | |
| <li>\(\sum_{i=1}^n a_i\) \(a_1 + a_2 + \ldots + a_n\)</li> | |
| <li>\(\sqrt{a}\) the non-negative square root of \(a\), for \(a \in \mathbb{R}, a \geq 0\)</li> | |
| <li>\(\sqrt[n]{a}\) the (real) \(n\)th root of \(a\), for \(a \in \mathbb{R}\), where \(n\) is an integer and \(a \geq 0\)</li> | |
| <li>\(|a|\) the modulus of \(a\)</li> | |
| <li>\(n!\) \(n\) factorial</li> | |
| <li>\(\binom{n}{r}\) the binomial coefficient \(\frac{n!}{r!(n-r)!}\) for \(n, r \in \mathbb{Z}\) and \(0 \leq r \leq n\)</li> | |
| </ul> | |
| <h2>4. Functions</h2> | |
| <ul> | |
| <li>\(f(x)\) the value of the function \(f\) at \(x\)</li> | |
| <li>\(f : A \to B\) \(f\) is a function under which each element of set \(A\) has an image in set \(B\)</li> | |
| <li>\(f : x \mapsto y\) the function \(f\) maps the element \(x\) to the element \(y\)</li> | |
| <li>\(f^{-1}\) the inverse function of the one-one function \(f\)</li> | |
| <li>\(gf\) the composite function of \(f\) and \(g\) which is defined by \(gf(x) = g(f(x))\)</li> | |
| <li>\(\lim_{x \to a} f(x)\) the limit of \(f(x)\) as \(x\) tends to \(a\)</li> | |
| <li>\(\Delta x, \delta x\) an increment of \(x\)</li> | |
| <li>\(\frac{dy}{dx}\) the derivative of \(y\) with respect to \(x\)</li> | |
| <li>\(\frac{d^n y}{dx^n}\) the \(n\)th derivative of \(y\) with respect to \(x\)</li> | |
| <li>\(f'(x), f''(x), \ldots, f^{(n)}(x)\) the first, second, \ldots, \(n\)th derivatives of \(f(x)\) with respect to \(x\)</li> | |
| <li>\(\int y \, dx\) the indefinite integral of \(y\) with respect to \(x\)</li> | |
| <li>\(\int_a^b y \, dx\) the definite integral of \(y\) with respect to \(x\) between the limits \(x = a\) and \(x = b\)</li> | |
| <li>\(\dot{x}, \ddot{x}, \ldots, \) the first, second, \ldots, derivatives of \(x\) with respect to \(t\)</li> | |
| </ul> | |
| <h2>5. Exponential and logarithmic functions</h2> | |
| <ul> | |
| <li>\(e\) base of natural logarithms</li> | |
| <li>\(e^x, \exp(x)\) exponential function of \(x\)</li> | |
| <li>\(\log_a x\) logarithm to the base \(a\) of \(x\)</li> | |
| <li>\(\ln x\) natural logarithm of \(x\)</li> | |
| <li>\(\lg x, \log_{10} x\) logarithm of \(x\) to base 10</li> | |
| </ul> | |
| <h2>6. Circular and hyperbolic functions</h2> | |
| <ul> | |
| <li>\(\sin, \cos, \tan, \csc, \sec, \cot\) the circular functions</li> | |
| <li>\(\sin^{-1}, \cos^{-1}, \tan^{-1}, \csc^{-1}, \sec^{-1}, \cot^{-1}\) the inverse circular functions</li> | |
| <li>\(\sinh, \cosh, \tanh, \csch, \sech, \coth\) the hyperbolic functions</li> | |
| <li>\(\sinh^{-1}, \cosh^{-1}, \tanh^{-1}, \csch^{-1}, \sech^{-1}, \coth^{-1}\) the inverse hyperbolic functions</li> | |
| </ul> | |
| <h2>7. Complex numbers</h2> | |
| <ul> | |
| <li>\(i\) the imaginary unit, \(i^2 = -1\)</li> | |
| <li>\(z\) a complex number, \(z = x + iy = r(\cos \theta + i \sin \theta)\)</li> | |
| <li>\(\Re z\) the real part of \(z\), \(\Re z = x\)</li> | |
| <li>\(\Im z\) the imaginary part of \(z\), \(\Im z = y\)</li> | |
| <li>\(|z|\) the modulus of \(z\), \(\sqrt{x^2 + y^2}\)</li> | |
| <li>\(\arg z\) the argument of \(z\), \(\arg z = \theta\) where \(-\pi < \theta \leq \pi\)</li> | |
| <li>\(z^*\) the complex conjugate of \(z\), \(x - iy\)</li> | |
| </ul> | |
| <h2>8. Matrices</h2> | |
| <ul> | |
| <li>\(\mathbf{M}\) a matrix \(\mathbf{M}\)</li> | |
| <li>\(\mathbf{M}^{-1}\) the inverse of the non-singular square matrix \(\mathbf{M}\)</li> | |
| <li>\(\det \mathbf{M}, |\mathbf{M}|\) the determinant of the square matrix \(\mathbf{M}\)</li> | |
| <li>\(\mathbf{I}\) an identity (or unit) matrix</li> | |
| </ul> | |
| <h2>9. Vectors</h2> | |
| <ul> | |
| <li>\(\mathbf{a}\) the vector \(\mathbf{a}\)</li> | |
| <li>\(\overrightarrow{AB}\) the vector represented in magnitude and direction by the directed line segment \(\overrightarrow{AB}\)</li> | |
| <li>\(\hat{a}\) a unit vector in the direction of \(\mathbf{a}\)</li> | |
| <li>\(\mathbf{i}, \mathbf{j}, \mathbf{k}\) unit vectors in the directions of the Cartesian coordinate axes</li> | |
| <li>\(\begin{pmatrix} x \\ y \\ z \end{pmatrix}\) the vectors \(x\mathbf{i} + y\mathbf{j}\) (in 2 dimensions) and \(x\mathbf{i} + y\mathbf{j} + z\mathbf{k}\) (in 3 dimensions)</li> | |
| <li>\(|\mathbf{a}|, \mathbf{a}\) the magnitude of \(\mathbf{a}\)</li> | |
| <li>\(|\overrightarrow{AB}|, \overrightarrow{AB}\) the magnitude of \(\overrightarrow{AB}\)</li> | |
| <li>\(\mathbf{a} \cdot \mathbf{b}\) the scalar product of \(\mathbf{a}\) and \(\mathbf{b}\)</li> | |
| <li>\(\mathbf{a} \times \mathbf{b}\) the vector product of \(\mathbf{a}\) and \(\mathbf{b}\)</li> | |
| </ul> | |
| <h2>10. Probability and statistics</h2> | |
| <ul> | |
| <li>\(A, B, C, \ldots\) events</li> | |
| <li>\(A \cup B\) union of the events \(A\) and \(B\)</li> | |
| <li>\(A \cap B\) intersection of the events \(A\) and \(B\)</li> | |
| <li>\(P(A)\) probability of the event \(A\)</li> | |
| <li>\(A'\) complement of the event \(A\)</li> | |
| <li>\(P(A | B)\) probability of the event \(A\) conditional on the event \(B\)</li> | |
| <li>\(\binom{n}{r}\) the number of combinations of \(r\) objects from \(n\), \(\binom{n}{r} = \frac{n!}{r!(n-r)!}\)</li> | |
| <li>\(\frac{n!}{(n-r)!}\) the number of permutations of \(r\) objects from \(n\)</li> | |
| <li>\(X, Y, R, \ldots\) random variables</li> | |
| <li>\(x, y, r, \ldots\) values of the random variables \(X, Y, R, \ldots\)</li> | |
| <li>\(x_1, x_2, \ldots\) observations</li> | |
| <li>\(f_1, f_2, \ldots\) frequencies with which the observations \(x_1, x_2, \ldots\) occur</li> | |
| <li>\(p(x)\) probability function \(P(X = x)\) of the discrete random variable \(X\)</li> | |
| <li>\(p_1, p_2, \ldots\) probabilities of the values \(x_1, x_2, \ldots\) of the discrete random variable \(X\)</li> | |
| <li>\(f(x)\) value of the probability density function of a continuous random variable \(X\)</li> | |
| <li>\(F(x)\) value of the cumulative distribution function of a continuous random variable \(X\)</li> | |
| <li>\(E(X)\) expectation of the random variable \(X\)</li> | |
| <li>\(E(g(X))\) expectation of \(g(X)\)</li> | |
| <li>\(\text{Var}(X)\) variance of the random variable \(X\)</li> | |
| <li>\(G_X(t)\) probability generating function for the discrete random variable \(X\)</li> | |
| <li>\(M_X(t)\) moment generating function for the random variable \(X\)</li> | |
| <li>\(B(n, p)\) binomial distribution with parameters \(n\) and \(p\)</li> | |
| <li>\(\text{Geo}(p)\) geometric distribution with parameter \(p\)</li> | |
| <li>\(\text{Po}(\lambda)\) Poisson distribution with parameter \(\lambda\)</li> | |
| <li>\(N(\mu, \sigma^2)\) normal distribution with mean \(\mu\) and variance \(\sigma^2\)</li> | |
| <li>\(\mu\) population mean</li> | |
| <li>\(\sigma^2\) population variance</li> | |
| <li>\(\sigma\) population standard deviation</li> | |
| <li>\(\bar{x}\) sample mean, \(\frac{1}{n} \sum_{i=1}^n x_i\)</li> | |
| <li>\(s^2\) unbiased estimate of population variance from a sample, \(\frac{1}{n-1} \sum_{i=1}^n (x_i - \bar{x})^2\)</li> | |
| <li>\(\rho\) product moment correlation coefficient for a population</li> | |
| <li>\(r\) product moment correlation coefficient for a sample</li> | |
| <li>\(\phi\) probability density function of the standardised normal variable \(Z \sim N(0, 1)\)</li> | |
| <li>\(\Phi\) cumulative distribution function of the standardised normal variable \(Z \sim N(0, 1)\)</li> | |
| <li>\(H_0, H_1\) null and alternative hypotheses for a hypothesis test</li> | |
| </ul> | |
| </body> | |
| </html> |
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