The cardinality of a set is a measure of its size, the number of its elements. Two sets have the same cardinality exactly when they are equinumerous.

To find the limit of $(2n + 1)/(n + 3)$, divide the numerator and the denominator by $n$ to get $(2 + 1/n)/(1 + 3/n)$. Since $1/n \to 0$, the rules give the limit $(2 + 0)/(1 + 0) = 2$.

A holomorphic map from the unit disc to itself that keeps $0$ fixed cannot move any point farther away from $0$. The only maps that keep the distance to $0$ of some point are the rotations.

Every symmetric polynomial can be written in exactly one way using the elementary symmetric polynomials. The coefficients stay in the same ring.

Take a function between subsets of Euclidean spaces, and a point $a$. The function is continuous at this point when small changes of the input near $a$ give small changes of the value. For a function of several variables, the input can change in all coordinates at once.

Take $p = 5$ and $a = 2$: $2^4 = 16 = 3 \cdot 5 + 1$, so $2^4 \equiv 1 \pmod 5$. The theorem says that this always happens for a prime $p$ and a number $a$ that $p$ does not divide. It lets you reduce large powers: $2^{100} = (2^4)^{25} \equiv 1 \pmod 5$.

A free module is a module that has a basis, that is, a linearly independent generating set.

Adding $U$ to $W$ adds the same new directions as $U$ has beyond its common part with $W$. The theorem makes this precise with quotient spaces. It is the linear algebra form of a general algebraic theorem.

The terms of a Cauchy sequence get closer and closer to each other. You can check this without knowing a limit, and this is the point. A Cauchy sequence looks as if it converges. But the space can have no point for it to converge to.

Fibonacci numbers are integers, but this formula uses $\sqrt{5}$, and the irrational parts cancel. The term with $\psi^n$ is very small, so $F_n$ is close to $\varphi^n / \sqrt{5}$.

Multiplicative functions stay multiplicative when you convolve them. So a new function built from known ones by divisor sums keeps the property, and you can compute it from prime powers.

Between each number and its double, there is always at least one prime. So the gap after a prime $p$ to the next prime is always smaller than $p$.

Negation is the logical operation that changes a proposition into its opposite. The negation of a true proposition is false, and the negation of a false proposition is true.

The step function is off before time $0$ and on after it. It models a switch, such as a voltage turned on at one instant.

The graph of $x^n$ starts at $0$ and grows past $y$. Since the graph has no gaps, it reaches the height $y$ at some point, which is the root.

The subgroup generated by a subset of a group is the smallest subgroup that contains the subset. Its elements are the finite products of elements of the subset and their inverses.

When we exchange two rows, the determinant changes its sign. A general reordering of the rows is a sequence of such exchanges. So it multiplies the determinant by $1$ or by $-1$, according to the sign of the permutation.

Near $a$, a function is close to its Taylor polynomial. The error is small even compared with $(x - a)^k$.

The image of a group homomorphism $\varphi \colon G \to H$ is the set of the values $\varphi(g)$ for $g \in G$. It is a subgroup of $H$.

The dual map is the linear algebra form of the transpose of a matrix. In dual bases, its matrix is the transpose of the matrix of the original map. This explains why rows and columns play symmetric roles.