Glossary of maths symbols

$$ \neg A $$

$$ not \ A $$

$$ A \lor B$$

$$ A \ or \ B $$

$$ A \land B$$

$$ A \ and \ B $$

$$ A \Longrightarrow B $$

$$ A \ implies \ B $$

$$ (If \ A , \ then \ B) $$

$$ B \Longrightarrow A $$

$$ reciprocal \ of \ (A \Longrightarrow B )$$

$$ A \Longrightarrow \neg B $$

$$ negation \ of \ (A \Longrightarrow B )$$

$$ \neg B \Longrightarrow \neg A $$

$$ contraposition \ of \ (A \Longrightarrow B) $$

$$ A \Longleftrightarrow B $$

$$ equivalence \ between \ A \ and \ B $$

$$ (A \ if \ and \ only \ if \ B) $$

$$ 0 $$

$$ 1 $$

$$ 1 $$

$$ 0 $$

$$ 0 $$

$$ 1 $$

$$ 1 $$

$$ 1 $$

$$ 0 $$

$$ 1 $$

$$ 0 $$

$$ 1 $$

$$ 0 $$

$$ 1 $$

$$ 0 $$

$$ 0 $$

$$ 1 $$

$$ 0 $$

$$ 1 $$

$$ 1 $$

$$ 0 $$

$$ 0 $$

$$ 1 $$

$$ 0 $$

$$ 1 $$

$$ 0 $$

$$ 0 $$

$$ 1 $$

$$ 1 $$

$$ 1 $$

$$ 1 $$

$$ 1 $$

$$ I = [a, b] $$

Interval from \(a\) to \(b\) including \(a\) and \(b\) (closed interval)

$$ I = ]a, b[ $$

Interval from \(a\) to \(b\) excluding \(a\) and \(b\) (open interval)

$$ \forall x \in I $$

For all \(x\) belonging to the interval \(I\)

$$ \forall (a, b) \in \hspace{0.05em} \mathbb{S}^2 $$

For any couple of numbers \((a, b)\), belonging to the \(\mathbb{S}\) set

$$ \exists x \in \mathbb{S}, \ A $$

It exists at least one number \(x\) inside the \(\mathbb{S}\) set, such as \(A\) is true

$$ \exists! x \in \mathbb{S}, \ A $$

It exists only one number \(x\) inside the \(\mathbb{S}\) set, such as \(A\) is true

$$ \mathbb{N} $$

Set of natural numbers: \( \bigl \{0, 1, 2, ..., n \bigr \}\)

$$ [\![ a, n]\!] $$

Set of natural numbers from \(a \) until \(n \): \( \bigl \{a, (a +1), (a + 2), ..., n \bigr \}\)

$$ [\![ 1, n]\!] = \{1, 2, ..., n\} $$

$$ \mathbb{Z} $$

Set of integers: \( \bigl \{-n, ..., -2, -1, 0, 1, 2, ..., n \bigr \}\)

$$ \mathbb{D} $$

Set of decimal numbers. All numbers that can be written in the form:

$$ d = \frac{a}{10^b} \hspace{3em} (a \in \mathbb{Z}, \ b \in \mathbb{N}) $$

$$ \mathbb{Q} $$

Set of rational numbers. All numbers that can be written in the form:

$$ r = \frac{p}{q} \hspace{3em} (p \in \mathbb{Z}, \ q \in \hspace{0.05em} \mathbb{N}^*) $$

$$ \mathbb{R} $$

Set of real numbers. All rational numbers \( (\in \mathbb{Q}) \) and irrational numbers \( (e, \ \pi, \ \phi, \ \sqrt{2} ...etc.) \).

$$ \mathbb{R} \backslash \Bigl \{a, b, c \Bigr \} $$

Set of real numbers deprived of values \(a, b, c \).

$$ \mathbb{C} $$

Set of complex numbers. All numbers that can be written in the form:

$$ c = a + ib \hspace{3em} ( (a,b) \in \hspace{0.05em} \mathbb{R}^2) $$

$$ \mathbb{K} $$

Set of scalars. All numbers belonging to \( \mathbb{R} \) or \( \mathbb{C} \)

$$ \mathbb{P} $$

Set of prime numbers. The set of numbers which have only themselves and \( 1\) as divisors:

$$ \mathbb{P} = \Bigl \{2, 3, 5, 7, 11, 13, ...etc. \Bigr\}$$

$$ \mathbb{K}^{ \mathbb{N}} $$

Set of real-valued or complex sequences

$$ D_f $$

Function \(f\) definition set

$$ f(x) = x^2$$

Definition of a function \(f\) having as variable \(x\) and as image \(x^2\)

$$ f :x \longmapsto x^2 $$

Definition of a function \(f\) having as variable \(x\) and as image \(x^2\)

$$ (f \circ g) (x) $$

Definition of a composite function \(f \circ g\) (read "\(f \) round \(g \)") having as variable \(x\)

$$(f \circ g) (x) = f\bigl(g(x)\bigr) $$

We can as well create composites of composites functions:

$$(f \circ g \circ h ) (x) = f\Bigl(g\bigl(h(x)\bigr)\Bigr) $$

$$...etc. $$

$$ \Biggl( \overset{n}{\underset{k=1}{\bigcirc f_k}} \ \Biggr ) (x) $$

$$ (non-official \ symbol \ a \ priori) $$

Composite functions operator:

$$ \Biggl( \overset{n}{\underset{k=1}{\bigcirc f_k}} \ \Biggr ) (x) = \Bigl(f_1 \circ f_2 \circ f_3 \circ \ ... \ \circ f_{n-1} \circ f_{n}\Bigr)(x) $$

$$ F(\mathbb{R}, \mathbb{R}) $$

Set of functions ranging from \(\mathbb{R}\) in \(\mathbb{R}\)

$$ f' $$

Derivative of the function \(f\) (Lagrange's notation)

$$ \frac{df}{dx} $$

Derivative of the function \(f\) in relation to \(x\) (Leibniz's notation)

$$ f^{(n)}$$

\(n\)-th derivative of the function \(f\)

$$ function \ of \ class \ \mathcal {C}^n \ on \ I $$

Continuous and \(n\)-times derivable function on the interval \(I\)

$$ \frac{\partial y}{\partial x} dx $$

Partial derivative of a function \(y = f(x, z,t) \) of several variables including \(x\), in relation to \(x\) uniquely.

$$ dy = \frac{\partial y}{\partial x} dx + \frac{\partial y}{\partial z}dz + \frac{\partial y}{\partial t}dt $$

Derivative of a function \(y = f(x, z, t ) \) having independent variables \(x, z, t \). It is the sum of all partial derivatives with respect to each variable respectively.

$$ TS_n(a) $$

Taylor series or order \(n\) in the neighbourhood of \(x = a\) (often \(0\))

$$ \int^x f(t) \ dt $$

The family of antiderivatives of the function \(f\) up to a constant.

$$ \int_a^x f(t) \ dt $$

Definite integral from \(a\) to \(x\) of the function \(f\).

Graphically, this corresponds to the area under the curve of the function \(f\) in the interval \([a, x]\), and this is also the integral of \(f\) which vanishes at \(a\).

$$ \int_a^b f(t) \ dt $$

Integral from \(a\) to \(b\) of the function \(f\).

Graphically, this corresponds to the area under the curve of the function \(f\) in the interval \([a, b]\).

$$ \sum_{k=0}^n \ f(k) $$

Sum from \(0\) until \(n\) of the \(f(k)\):

$$ \sum_{k=0}^n \ f(k) = f(0) + f(1) + ... + f(n) $$

$$ \sum u_n $$

Numerical series associated with a sequence \( (u_n)_{n \in \mathbb{N}}\)

$$ S_n = \sum_{k=0}^n u_n $$

Partial sum of the series \( \sum u_n\), from \(0\) until \(n\)

$$ R_n = \sum_{k=n+1}^{+ \infty} u_n $$

Remainder of the series \( \sum u_n\):

$$ \sum u_n = S_n + R_n = \sum_{k=0}^n u_n + \sum_{k=n+1}^{+ \infty} u_n $$

$$ \prod_{k= 0}^n \ f(k) $$

Product from \(0\) until \(n\) of the \(f(k)\):

$$ \prod_{k=0}^n \ f(k) = f(0) f(1) ... f(n) $$

$$ lim_{x \to a} \ f(x) = l $$

Limit of a function \(f \) when \(x \) tends towards \( a \).

\( a \) and \( l \) can be a number or an extremity \( (-\infty\) or \(+\infty) \) .

$$ f(x) \underset{a}{\longmapsto} l $$

Limit of a function \(f \) when \(x \) tend towards \( a \) (shortened writing)

We read: "\( f(x) \) tends towards \( l \), when \(x \) tends towards \( a \)".

$$ n! $$

Factorial of \(n \):

$$n! = n \times (n-1) \times (n-2) \ \times \ ... \ \times \ 2 \times 1$$

$$ \binom{n}{k} $$

The number of ways to pick \( k \) elements among \(n \).

We read: "\( k \) among \(n \)".

$$\binom{n}{k} = \frac{n !}{(n-k)! \ k !}$$

$$ M = \begin{pmatrix} x_{1,1} & x_{1,2} & x_{1,3} & \dots & x_{1, m} \\ x_{2,1} & x_{2,2} & x_{2,3} & \dots & x_{2, m} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ x_{n,1} & x_{n,2} & x_{n,3} & \dots & x_{n, m} \\ \end{pmatrix} $$

The matrix \( M \) having \( n \) lines and \( p \) columns.

$$ M_3 = \ \begin{pmatrix} x_{1,1} & x_{1,2} & x_{1,3} \\ x_{2,1} & x_{2,2} & x_{2,3} \\ x_{3,1} & x_{3,2} & x_{3,3} \\ \end{pmatrix} $$

The sqaure matrix \( M_3 \) having \( 3 \) lines and \( 3 \) columns.

$$ \underbrace{ \begin{pmatrix} x_{1,1} & x_{1,2} & x_{1,3} & \dots & x_{1, p} \\ x_{2,1} & x_{2,2} & x_{2,3} & \dots & x_{2, p} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ x_{n,1} & x_{n,2} & x_{n,3} & \dots & x_{n, p} \\ \end{pmatrix} } _\text{M} \times \underbrace{ \begin{pmatrix} a_0 \\ a_1 \\ a_2 \\ . \\ a_n \\ \end{pmatrix} } _\text{A} = \underbrace{ \begin{pmatrix} y_0 \\ y_1 \\ y_2 \\ . \\ y_n \\ \end{pmatrix} } _\text{Y} $$

The matrix multiplication between the matrix \( M\) and the matrix \( A\).

The number of columns of the matrix \( M\) must be the same as the number of lines of the matrix \( A\). The result of it, the matrix \( Y\), is a matrix of the same nature than the one positionned at the right of the product.

$$ det(M) = \begin{vmatrix} x_{1,1} & x_{1,2} & x_{1,3} & \dots & x_{1, p} \\ x_{2,1} & x_{2,2} & x_{2,3} & \dots & x_{2, p} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ x_{n,1} & x_{n,2} & x_{n,3} & \dots & x_{n, p} \\ \end{vmatrix} $$

The determinant of the matrix \( M \) having \( n \) lines and \( p \) columns.

$$ A \in \hspace{0.03em} \mathcal{M}_{n,p} (\mathbb{K})$$

The matrix \(A\) belonging to set of all matrix (and more precisely the vector space) having \(n\) lines and \(p\) columns on the field \(\mathbb{K}\).

$$ A \in \hspace{0.03em} \mathcal{M}_{n} (\mathbb{K})$$

The squared matrix \(A\) belonging to set of all squared matrix (and more precisely the vector space) of the size \(n\) on the field \(\mathbb{K}\).

$$ \mathcal{D}(a)$$

The set of divisors of \(a\)

$$ p \in \mathbb{P}$$

\(p\) is a prime number

$$ \mathcal{D}(p) = \{1, p\}$$

\(p\) is a prime number

$$ a / b$$

\(a\) divides \(b\)

$$ a \nmid b $$

\(a\) does not divide \(b\)

$$ \mathcal{D}(a, b) $$

The set of common divisors of \(a\) and \(b\)

$$ \delta = GCD(a, b)= a \wedge b $$

\( \delta\) is the greatest common divisor of \(a\) and \(b\).

$$ a \wedge b = 1 $$

\(a\) and \(b\) are coprime. We can also say that they are "foreigners".

$$ LCM(a, b) $$

\( LCM(a, b) \) is the lowest common multiple of \(a\) and \(b\).

$$ a \equiv b \hspace{0.2em} [n] $$

\(a\) and \(b\) are congruent modulo \(n\).

We say that \(a\) and \(b\) are congruent modulo \(n\) f they have the same remainder \(R\) in the Euclidian division by \(n\).

$$ a \equiv b \hspace{0.2em} [n] \Longleftrightarrow \exists (q, q') \in \hspace{0.05em} \mathbb{N}^2, \enspace \exists R \in \hspace{0.05em} \mathbb{N}, \enspace 0 \leqslant R < n, \ \Biggl \{ \begin{align*} a = nq + R \\ b= nq' + R \end{align*} $$

$$ \vec{u} $$

A vector \(\vec{u} \)

$$ \vec{u}\begin{pmatrix} x\\ y\\ z \end{pmatrix} $$

A vector \(\vec{u} \) having as coordinates \(x, y ,z \)

$$ || \vec{u} || $$

The norm (or length) of a vector \( \vec{u}\).

Let be a vector \(\vec{u} \begin{pmatrix} x\\ y\\z \end{pmatrix}\), then its norm is worth:

$$ || \vec{u} || = \sqrt{x^2 + y^2 + z^2}$$

$$ \vec{u}. \vec{v}$$

The scalar product of two vectors \( \vec{u}\) and \( \vec{v}\) is worth:

$$ \vec{u}. \vec{v} = || \vec{u} || \times || \vec{v} || \times cos( (\vec{u}, \vec{v}) ) $$

Otherwise, depending on the coordinates, if we have two vectors \(\vec{u}\begin{pmatrix} x\\ y\\z \end{pmatrix}\) and \(\vec{v} \begin{pmatrix} x'\\ y'\\z' \end{pmatrix}\), then this scalar product is worth:

$$ \vec{u}. \vec{v} = xx' + yy' +zz' $$

$$ \vec{u} \land \vec{v}$$

The vector product of vectors \(\vec{u}\begin{pmatrix} x\\ y\\z \end{pmatrix}\) and \(\vec{v} \begin{pmatrix} x'\\ y'\\z' \end{pmatrix}\) is worth:

$$ \vec{u} \land \vec{v} = \begin{pmatrix} y_1.z_2 - y_2.z_1 \\ x_2.z_1 - x_1.z_2 \\ x_1.y_2 - x_2.y_1 \end{pmatrix} $$

$$ alpha $$

$$ \alpha $$

$$ A $$

$$ b\textit{ê}ta $$

$$ \beta $$

$$ B $$

$$ gamma $$

$$ \gamma $$

$$ \Gamma $$

$$ delta $$

$$ \delta $$

Difference between two physically measurable elements.

Examples:

1. calculating a slope between two points \(A\) and \(B\):

$$ m = \frac{\Delta y}{\Delta x} = \frac{ y_b - y_a }{ x_b - x_a }$$

2. calculation of the discriminant in solving quadratic equations:

$$ \Delta = b^2 - 4ac$$

$$ epsilon $$

$$ \varepsilon $$

$$ E $$

$$ z \textit{ê}ta $$

$$ \zeta $$

$$ Z $$

$$ \textit{ê}ta $$

$$ \eta $$

$$ H $$

$$ th\textit{ê}ta $$

$$ \theta $$

$$ \Theta $$

$$ iota $$

$$ \iota $$

$$ I $$

$$ kappa $$

$$ \kappa $$

$$ K $$

$$ lambda $$

$$ \lambda $$

$$ \Lambda $$

$$ mu $$

$$ \mu $$

$$ M $$

$$ nu $$

$$ \nu $$

$$ N $$

$$ xi $$

$$ \xi $$

$$ \Xi $$

$$ omicron $$

$$ o $$

$$ O $$

$$ pi $$

$$ \pi $$

$$ \Pi $$

$$ rho $$

$$ \rho $$

$$ P $$

$$ sigma $$

$$ \sigma $$

$$ \Sigma $$

$$ tau $$

$$ \tau $$

$$ T $$

$$ upsilon $$

$$ \upsilon $$

$$ \Upsilon $$

$$ phi $$

$$ \phi $$

$$ \Phi $$

$$ chi \ (pronounced "ki") $$

$$ \chi $$

$$ X $$

$$ psi $$

$$ \psi $$

$$ \Psi $$

$$ omega $$

$$ \omega $$

$$ \Omega $$