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Let $a < b$ and let $f \colon [a, b] \to \mathbb{R}$ be monotone. Then $f$ is Riemann integrable on $[a, b]$.

Let $(a_n)$ and $(b_n)$ be real sequences with $0 \le a_n \le b_n$ for all $n$. If $\sum b_n$ converges, then $\sum a_n$ converges, and $\sum a_n \le \sum b_n$. If $\sum a_n$ diverges, then $\sum b_n$ diverges.

A tautology is a formula of propositional logic that is true for every assignment of truth values to its variables.

A harmonic conjugate of a real harmonic function $u$ is a real function $v$ such that $u + iv$ is holomorphic, see the definition. On a domain, it is unique up to an additive constant when it exists.

The symmetric group on a set $X$ is the group of all bijections from $X$ to $X$, with composition as the operation. For $X = \{1, \dots, n\}$ it is written $S_n$, and its elements are the permutations of $n$ symbols.

The set $\mathbb{R}$ of real numbers is uncountable.

Suppose that all cases are equally likely. Then the classical definition says that the probability of an event is the number of favourable cases divided by the number of all cases. It goes back to Jacob Bernoulli and Pierre-Simon Laplace. The node classical definition of probability gives the precise form.

Let $p$ be a prime number, and let $a$ and $b$ be integers. If $p \mid a b$, then $p \mid a$ or $p \mid b$.

Let $K$ be a commutative ring, let $n \ge 2$, and let $A$ be an $n \times n$ matrix over $K$. Then $$A \operatorname{adj}(A) = \operatorname{adj}(A) A = \det(A)\, I_n,$$ where $\operatorname{adj}(A)$ is the adjugate of $A$ and $\det$ is the determinant.

A graph homomorphism is a map between the vertex sets of two graphs that preserves adjacency. Homomorphisms unify colourings and other graph maps. The node graph homomorphism gives the precise definition.

Let $F$ be a field, let $V$ be a finite-dimensional vector space over $F$, and let $T \colon V \to V$ be a linear map. Let $p_T$ be the characteristic polynomial of $T$, and let $\lambda \in F$. Then $\lambda$ is an eigenvalue of $T$ if and only if $p_T(\lambda) = 0$.

The converse of a relation is the relation with all pairs reversed.

Let $(X, d)$ be a metric space and let $K \subseteq X$. Then $K$ is compact if and only if $K$, with the subspace metric, is complete and $K$ is totally bounded.

The dihedral group $D_4$ is the group of the symmetries of a square, and it has order $8$. Let $r$ be the rotation by $90$ degrees and let $s$ be a reflection in a line of symmetry. Then $D_4 = \{e, r, r^2, r^3, s, rs, r^2s, r^3s\}$, with $r^4 = e$, $s^2 = e$ and $srs^{-1} = r^{-1}$. The elements $r$ and $r^3$ have order $4$, and the other five elements different from $e$ have order $2$. The centre of $D_4$ is $\{e, r^2\}$, because the rotation by $180$ degrees commutes with each symmetry. The rotation subgroup $\langle r \rangle$ is normal of index $2$. The subgroup $\{e, s\}$ is not normal, because $r s r^{-1} = r^2 s$. Numbering the vertices $1, 2, 3, 4$ gives an injective homomorphism from $D_4$ into $S_4$, and its image is a Sylow $2$-subgroup of $S_4$.

A sesquilinear form on a complex vector space is a map of two vectors to the complex numbers. It is linear in one argument and conjugate linear in the other.

Let $m$ and $n$ be distinct natural numbers. Then the Fermat numbers $F_m$ and $F_n$ are coprime.

Let $R$ be a principal ideal domain. Then $R$ is a unique factorisation domain.

The perpendicular bisector of a segment passes through the midpoint of the segment at a right angle to it. In the plane it is a line, and in space it is a plane. Its points are exactly the points at the same distance from the two endpoints.

A function is sequentially continuous when it maps each convergent sequence in its domain to a sequence that converges to the value at the limit. For real functions, this is the same as continuity. The node sequential continuity gives the precise definition.

Let $a < b$ and let $f \colon [a, b] \to \mathbb{R}$ be bounded. Then $f$ is Riemann integrable if and only if the set of points where $f$ is not continuous has Lebesgue measure zero.

Let $F$ be a field and let $A$ be an $n \times n$ matrix over $F$. Then $A$ is diagonalisable if and only if $F^n$ has a basis of eigenvectors of $A$. If $v_1, \dots, v_n$ is such a basis with $A v_i = \lambda_i v_i$, and $P$ has the columns $v_1, \dots, v_n$, then $P^{-1} A P$ is the diagonal matrix with entries $\lambda_1, \dots, \lambda_n$.

Let $ABC$ be a triangle in a Euclidean space $\mathbb{R}^n$ with $d(A, B) = d(A, C)$, where $d$ is the Euclidean distance. Then the angles at $B$ and $C$ are equal: $\angle ABC = \angle ACB$.

Let $p(n)$ be the partition function. Then, as formal power series in $\mathbb{Z}[\![x]\!]$, $$\sum_{n \ge 0} p(n)\, x^n = \prod_{k \ge 1} \frac{1}{1 - x^k}.$$

Let $n$ be a positive integer. A complex number $z$ is an $n$th root of unity if and only if $$z = e^{2 \pi i k / n} = \cos \frac{2 \pi k}{n} + i \sin \frac{2 \pi k}{n}$$ for some integer $k$ with $0 \le k < n$.

Let $n$ be a positive integer. Then the number of compositions of $n$ is $2^{n-1}$.

Let $A$, $B$, $C$ and $D$ be concyclic points of the Euclidean plane $\mathbb{R}^2$, and let $d$ be the Euclidean distance. Suppose that the segments $AC$ and $BD$ meet in a point that is inside both segments. Thus $ABCD$ is a convex cyclic quadrilateral with the diagonals $AC$ and $BD$. Then $$d(A, C)\, d(B, D) = d(A, B)\, d(C, D) + d(B, C)\, d(A, D).$$

Let $a = (a_1, a_2, a_3)$, $b = (b_1, b_2, b_3)$ and $c = (c_1, c_2, c_3)$ be vectors in $\mathbb{R}^3$. Let $\times$ be the cross product and let $\cdot$ be the dot product. The scalar triple product of $a$, $b$ and $c$ is the real number $a \cdot (b \times c)$. It is equal to the determinant of the matrix whose rows are $a$, $b$ and $c$: $$a \cdot (b \times c) = \det \begin{pmatrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \end{pmatrix}.$$

Let $F$ be a field, let $V$ be a vector space over $F$, and let $B = (b_1, \dots, b_n)$ be a basis of $V$. Then the map $$V \to F^n, \qquad v \mapsto [v]_B,$$ that sends each vector to its coordinate vector in $B$, is a linear isomorphism. Its inverse sends $(a_1, \dots, a_n)$ to $a_1 b_1 + \cdots + a_n b_n$.

A probability measure is a countably additive function on a σ-algebra of events that gives the whole space the value $1$. It assigns to each event its probability. The node probability measure gives the precise definition.

A partition of a closed interval is a finite increasing list of points that starts at the left end and stops at the right end. It cuts the interval into subintervals. The node partition of an interval gives the precise definition.

Let $r$ be a real number with $r \ne 1$, and let $n$ be a natural number. Then $$\sum_{k=0}^{n-1} r^k = 1 + r + r^2 + \cdots + r^{n-1} = \frac{1 - r^n}{1 - r}.$$

The normal distribution with mean $\mu$ and variance $\sigma^2$ is the continuous distribution on $\mathbb{R}$ with the bell-shaped density $\frac{1}{\sqrt{2\pi\sigma^2}} e^{-(x-\mu)^2/(2\sigma^2)}$. By the central limit theorem, it is the approximate law of a sum of many small independent effects. The node normal distribution gives the precise definition.

A binary relation between two sets says, for each pair of elements, whether the first element is related to the second. Formally, it is a set of ordered pairs.

Let $X$ and $Y$ be independent integrable real random variables. Then $XY$ is integrable and $$E[XY] = E[X] \, E[Y].$$

Let a group $G$ act on a set $X$, let $x \in X$, and let $g \in G$. Then the stabiliser of $g \cdot x$ is $G_{g \cdot x} = g G_x g^{-1}$.

The tangent function is the quotient of sine by cosine. It has period $\pi$, and it has vertical asymptotes where the cosine vanishes. The node tangent function gives the precise definition.

The determinant of a family of $n$ vectors in a basis of an $n$-dimensional space is a scalar. It is the determinant of the matrix of their coordinates in that basis.

The boundary of a subset of a topological space is the set of points in its closure that are not in its interior. Each neighbourhood of a boundary point meets both the subset and its complement.

Let $F_k$ be the Fibonacci numbers, and let $N$ be a positive integer. Then there is a unique finite set $\{c_1, \dots, c_r\}$ of integers, with $r \ge 1$, such that $$N = F_{c_1} + F_{c_2} + \dots + F_{c_r},$$ each $c_i \ge 2$, and $c_{i+1} \ge c_i + 2$ for $1 \le i < r$.

Pascal's triangle is the triangular array of the binomial coefficients. Its rows are numbered $n = 0, 1, 2, \dots$ from the top. Row $n$ has $n + 1$ entries, $\binom{n}{0}, \binom{n}{1}, \dots, \binom{n}{n}$, from left to right.