If you have a chain of nonempty compact sets, each inside the one before, at least one point is in all of them. For example, the intervals $[0, 1/n]$ are nested, and $0$ is in all of them.

A branch of the logarithm on an open set $U \subseteq \mathbb{C} \setminus \{0\}$ is a continuous function $L$ on $U$ with $e^{L(z)} = z$, see the definition. On a domain, two branches differ by a constant integer multiple of $2\pi i$.

The Laplace transform is an integral transform. It sends a function $f$ of a real variable $t \ge 0$ to the function $F(s) = \int_0^\infty f(t) e^{-st}\, dt$ of a complex variable $s$. It is named after Pierre-Simon Laplace. See the definition.

The union of sets is the set of all objects that are elements of at least one of the sets.

The equation $a^4 + b^4 = c^4$ has no whole-number solutions. The proof assumes a solution and builds a smaller one from it, which leads to a contradiction.

A residue class modulo $n$ is the set of all integers that are congruent to a given integer modulo $n$. The definition of a residue class gives the precise form.

The conjugate transpose of a complex matrix is the transpose of the matrix with each entry replaced by its complex conjugate.

To choose $r$ people from two groups together, choose $k$ from the first group and $r - k$ from the second, for each possible $k$. Then add the counts for all values of $k$.

A continuous map is a function that sends nearby points to nearby points. Between topological spaces, this means that preimages of open sets are open. Between metric spaces, it is the same as the epsilon-delta condition.

Apply the contraction again and again, starting from any point. The points get closer to each other, and in a complete space they converge. The limit does not move under $f$, and it is the only point that does not move.

A tree is a connected graph with no cycles. Trees are the minimal connected graphs, and they appear in data structures, in spanning problems and in counting problems. The node tree gives the precise definition.

A T0 space, or Kolmogorov space, is a topological space in which the open sets can tell any two distinct points apart. It is the weakest of the usual separation axioms.

Draw three lines from the corners of a triangle through one common inside point. Each line cuts the opposite side into two parts. Multiply the three ratios of the parts, going round the triangle in one direction: the product is always $1$.

Add the sizes of the single sets, subtract the sizes of the pairwise intersections, add the sizes of the triple intersections, and so on. Each element of the union is then counted exactly once in total.

If we fix all rows but one, then the determinant is a linear function of the remaining row. So a sum in one row splits the determinant into a sum of two determinants. A factor in one row comes out of the determinant.

The stabiliser of a point $x$ under a group action is the set of the group elements that do not move $x$. It is a subgroup of the group.

A Boolean function is a function from $n$-tuples of bits to bits. Boolean functions are the objects that logic formulas and digital circuits compute. The node Boolean function gives the precise definition.

If you know all the divisor sums of $f$, the formula recovers $f$ itself. It undoes the operation "sum over the divisors", as subtraction undoes addition.

The number $-1$ is a square modulo an odd prime $p$ exactly when $p$ leaves remainder $1$ on division by $4$. So $-1$ is a square modulo $5$, $13$ and $17$, but not modulo $3$, $7$ and $11$.

This is the theorem of Pythagoras for vectors. Two orthogonal vectors are the legs of a right triangle, and their sum is the hypotenuse. It holds in every inner product space, even in spaces of functions.