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 "nth 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
$\sum_{i = q}^m i = i + (i + 1) + ... + (q-1) + q$ 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 $\binom{n}{r} = \frac{n!}{k!(n - k)!}$.