Let $(\Omega, \mathcal F, \mathbb P)$ be a probability space, and let $A,B \in \mathcal F$ with $\mathbb P(B)>0$. Then $A$ and $B$ are independent if and only if \begin{align*} \mathbb P(A \mid B)=\mathbb P(A). \end{align*} If also $\mathbb P(A)>0$, this is equivalent to \begin{align*} \mathbb P(B \mid A)=\mathbb P(B). \end{align*}

Let $X_1, \ldots, X_n$ be random variables taking values in countable sets $E_1, \ldots, E_n$. Then $X_1, \ldots, X_n$ are independent if and only if for every $(x_1, \ldots, x_n) \in E_1 \times \cdots \times E_n$, \begin{align*} \mathbb P(X_1=x_1, \ldots, X_n=x_n) = \prod_{i=1}^n \mathbb P(X_i=x_i). \end{align*}

Let $(V, (\cdot, \cdot)_V)$ be an [inner product space](/page/Inner%20Product%20Space). The function $\|\cdot\|_V$ defined by \begin{align*} \|x\|_V &= \sqrt{(x,x)_V} \end{align*} is a norm on $V$.

Let $(V, (\cdot, \cdot)_V)$ be an [inner product space](/page/Inner%20Product%20Space) and let $M \subset V$ be a finite-dimensional subspace. For each $x \in V$, there exists a unique $p \in M$ such that $x-p \in M^\perp$. Moreover, \begin{align*} \|x-p\|_V &= \inf_{m \in M} \|x-m\|_V. \end{align*} If $e_1, \ldots, e_m$ is an orthonormal basis of $M$, then \begin{align*} p &= \sum_{i=1}^m (x,e_i)_V e_i. \end{align*}

For every ordinal $\alpha$, the cofinality $\operatorname{cf}(\alpha)$ is a cardinal, where a cardinal is understood as an initial ordinal.

For all ordinals $\alpha$ and $\beta$, if $\alpha \leq \beta$, then $L_\alpha \subset L_\beta$, where the constructible hierarchy is defined by \begin{align*} L_0 &= \varnothing,\\ L_{\gamma+1} &= \operatorname{Def}(L_\gamma),\\ L_\lambda &= \bigcup_{\gamma < \lambda} L_\gamma \end{align*} for every ordinal $\gamma$ and every limit ordinal $\lambda$.

Let $\mathrm{Card}$ denote the class of cardinals, where each cardinal is represented by its initial ordinal. Define a relation $\le$ on $\mathrm{Card}$ by declaring, for $\kappa, \lambda \in \mathrm{Card}$, $\kappa \le \lambda$ iff there exists an injective map $f: \kappa \to \lambda$. Then $\le$ is a partial order on $\mathrm{Card}$: for all cardinals $\kappa, \lambda, \mu$, $\kappa \le \kappa$; if $\kappa \le \lambda$ and $\lambda \le \mu$, then $\kappa \le \mu$; and if $\kappa \le \lambda$ and $\lambda \le \kappa$, then $\kappa = \lambda$.

For every set $A$, there exists an initial ordinal $\kappa$ such that no injective map $f: \kappa \to A$ exists.

Let $\theta$ be a cardinal such that $\omega_1 \in H_\theta$. Let $M \preccurlyeq (H_\theta,\in)$ be a countable elementary submodel with $\omega_1 \in M$. Then $M$ is countable as a set in the ambient universe, but \begin{align*} (M,\in) \models \text{``there is no surjection from } \omega \text{ onto } \omega_1\text{''}. \end{align*} Equivalently, there is no $f \in M$ such that $M$ regards $f$ as a surjective map $f:\omega \to \omega_1$.

For all ordinals $\alpha$, $\beta$, and $\gamma$ with $\beta \leq \gamma$, the following inequalities hold: \begin{align*} \alpha + \beta &\leq \alpha + \gamma,\\ \alpha \cdot \beta &\leq \alpha \cdot \gamma. \end{align*} Moreover, if $\alpha > 0$, then \begin{align*} \alpha^\beta \leq \alpha^\gamma. \end{align*}