Glossary
Glossary of Mathematical Terms
Term | Meaning / Use |
---|---|
Lemma | Sometimes referred to a "small theorem" or "helping theorem". They are small assertions often unimportant by themselves but are used to prove much more advanced theorems. |
Theorem | An assertion built upon by lemmas. |
Corollary | A result of a lemma or theorem. |
Proposition | |
Axiom | An unprovable assertion or characteristic (e.g. between two points in space there exists a measurement of distance). |
Mathematical Notation
Numbers
Quantity |
---|
$\pi \approx 3.14159...$ (Pi) |
$e \approx 2.71828$ (Euler's number) |
$\phi \approx 1.61803...$ (The golden ratio) |
$i = \sqrt{-1}$ (The imaginary number) |
Calculus Notation
Symbol / Notation | Meaning / Use |
---|---|
$\lim_{x \to h} f(x)$ | Read as "the limit as the variable x approaches h". |
$\frac{dy}{dx} \: , \: f'(x)$ | Refers to the "derivative of y (or f(x)) with respect to x" , and denotes a function describing rate of change. |
$\frac{d^2y}{dx^2} \: , \: f''(x)$ | Refers to the "second derivative of y (or f(x)) with respect to x". |
$\frac{d^{n}y}{dx^{n}} \: , \: f^{(n)}(x)$ | Refers to the "n^{th} derivative of y (or f(x)) with respect to x". |
$\int y \: dx \: , \: \int f(x) \: dx$ | Refers to the "indefinite integral of y (or f(x)) with respect to x". |
$\int_a^b y \: dx \: , \: \int_a^b f(x) \: dx$ | Refers to the "definite integral bounded from x = a to x = b of y (or f(x)) with respect to x". |
Graph Theory Notation
Symbol / Notation | Meaning / Use |
---|---|
$V(G)$ | The vertex set of a graph G containing elements known as vertices. |
$E(G)$ | The edge set of a graph G containing elements known as edges where every edge $e_i = \{v_i, v_j \}$. |
$G \cong H$ | Isomorphism between the graphs G and H. Two graphs G and H are isomorphic is there exists a bijection $f: V(G) \to V(H)$ so that an edge appears in E(G) if and only if a corresponding edge appears in E(H). |
$\delta (G)$ | The minimum vertex degree in a graph G. |
$\Delta (G)$ | The maximum vertex degree in a graph G. |
$G^*$ | The dual of a planar graph where vertices in G become faces in G^{*} and vice versa. |
$\kappa (G)$ | The minimum vertex cutset. |
$\lambda (G)$ | The minimum edge cutset. |
$\chi (G)$ | The chromatic number of a graph G, that is the minimum number of colours for a good vertex k-colouring of G. |
$\chi ' (G)$ | The chromatic index of a graph G, that is the minimum number of colours for a good edge k-colouring of G. |
Linear Algebra
Symbol / Notation | Meaning / Use |
---|---|
$\mathbb{R}^n$ | Euclidean n-space. |
$\vec{u} = (u_1, u_2, ..., u_n)$ | Denotes a vector in Euclidean n-space. |
$\mid \mid \vec{u} \mid \mid$ | The norm of the vector (or length of the vector) u. |
$\vec{u} \perp \vec{v}$ | Reads "vector u is perpendicular to vector v". |
$\vec{i}, \: \vec{j}, \: \vec{k}$ | Standard unit vectors for Euclidean n-space. |
$\vec{u} \cdot \vec{v}$ | Represents the dot product between two vectors that produces a scalar quantity. |
$\vec{u} \times \vec{v}$ | Represents the cross product between two vectors that products another vector. The cross product is only defined in Euclidean 3-space. |
$proj_{\vec{b}} \vec{u}$ | Denotes the projection of vector u onto vector b. |
$T: \mathbb{R}^n \mapsto \mathbb{R}^n$ | A linear transformation from Euclidean n-space to Euclidean n-space. |
Number Theory Notation
Symbol / Notation | Meaning / Use |
---|---|
$a \mid b$ | Read as "a divides b" (a and b both integers) that is there exists an integer q such that $aq = b$. |
$(a, b) = d \: \mathbf{or} \: gcd(a, b) = d$ | The greatest common divisor of integers a and b. The integer d is such that d | a and d | b, and if c | a and c | b, c ≤ d. |
$a \equiv b \pmod m$ | Linear congruence. That is $m \mid (b - a)$ (a, b, m all integers). |
$d(n)$ | Number of positive divisors of an integer n. |
$\sigma (n)$ | Sum of all positive divisors of an integer n. |
$\phi (n)$ | Euler's totient function. Denotes the number of positive integers less than n that are relatively prime to n. |
$(a/p)$ | The Legendre symbol that determines whether the quadratic congruence $x^2 \equiv a \pmod p$ has solutions. |
Set Notation
Symbol / Notation | Meaning / Use |
---|---|
$\mathbb{R}$ | The set of real numbers. |
$\mathbb{C}$ | The set of complex numbers $z = a + bi$ where a is the "real part of z" and bi is the "imaginary part of z". |
$\mathbb{N}$ | The set of natural numbers 0, 1, 2, …. Some definitions do not state 0 as a natural number. |
$\mathbb{Q}$ | The set of rational numbers $q = \frac{a}{b}$ where a and b are both integers and b ≠ 0. |
$\mathbb{Z}$ | The set of integers: …, -2, -1, 0, 1, 2, … |
$x \in S$ | Set membership, that is the element x is an element of the set S. If the element x is not in set S, then $x \not \in S$. |
$A \wedge B$ | Read as "A and B". |
$A \vee B$ | Read as "A or B". |
$\forall x$ | Read as "for all x". |
$\exists x$ | Read as "there exists and x" |
$\mid \: A \: \mid$ | The cardinality of set A, or the number of elements in S. |
$A - B$ | Set subtraction A minus B, that is $A - B = \{ a \in A : a \not \in B \}$. |
$A \times B$ | The cartesian product of the sets A and B, that is $A \times B = \{(a, b) : a \in A, b \in B \}$. |
$A \cap B$ | The intersection of sets A and B, that is $A \cap B = \{ x : x \in A \wedge x \in B \}$ or the set of elements that are in both sets A and B. |
$A \cup B$ | The union of sets A and B, that is $A \cup B = \{ x : x \in A \vee x \in B \}$ or the set of elements that are in either set A, set B, or both set A and B. |
Other Notation
Symbol / Notation | Meaning / Use |
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$\sum_{i = q}^m i = i + (i + 1) + ... + (q-1) + q$ | The sum of all integers from i = q to i = m. |
$\prod_{i = q}^m i = i(i+1)...(q-1)q$ | The product of all integers from i = q to i = m. |
$n!$ | Read as "n factorial", and is defined such that $n! = n \cdot (n - 1) \cdot (n - 2) \cdot ... \cdot 2 \cdot 1$. Note that 0! = 1 by definition. |
$\binom{n}{r} \: , \: n C r$ | Read as "n choose r" and is defined such that $\binom{n}{r} = \frac{n!}{k!(n - k)!}$. |