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A set $S$ of formulae is deductively closed whenEquivalently, $S$ equals its deductive closure $\mathrm{Cn}(S)$.

If $\Gamma ⊢t$, every valuation that satisfies all formulae in $\Gamma$ also satisfies $t$.

This is soundness for the propositional proof system and valuation semantics.

Intuitionistic propositional logic omits unrestricted excluded middle and interprets implication constructively.

A uniformly accelerated observer has a null boundary beyond which light signals can never reach the observer.

A Möbius transformation is represented by an invertible two-by-two matrix, with nonzero scalar multiples representing the same map.

A Möbius transformation of the Riemann sphere has the formThe group is generated by translations, nonzero complex scalings, and inversion $z\mapsto 1/z$.

Over an algebraically closed field of characteristic other than two, the rank-$r+1$ quadricis an irreducible $r$-dimensional cone for $r\ge 2$, and its singular locus is the omitted-coordinate vertex $[0:\cdots :0:1]$.

If $X=V_{+}(F)\subseteq {\mathbb{P}}^n$, its affine cone is $Y=V(F)\subseteq {\mathbb{A}}^{n+1}$. If $X$ is smooth, every nonzero point of $Y$ is smooth; hence the vertex is the only possible singular point. The vertex is smooth for linear $F$ and singular whenever $\mathrm{deg}F\ge 2$.

Given homogeneous polynomials $F,G$ of the same degree, their pencil is the one-parameter familyIts total space is the projective hypersurface $V(sF+tG)\subseteq {\mathbb{P}}^2\times {\mathbb{P}}^1$, and projection to $[s:t]$ has these plane curves as its fibres.

It is a pencil whose members are plane curves.

For a projective algebraic set $X\subseteq {\mathbb{P}}^n$, its affine cone is the union in ${\mathbb{A}}^{n+1}$ of the lines represented by the points of $X$, together with the origin.

If $\Vert \nabla F(x_i)\Vert \le G$ and $\Vert x_1-x_{\ast }\Vert \le D$, thenIt follows by expanding the squared distance after each projected step, using nonexpansiveness of projection, summing the resulting inequalities, and applying convexity.

On a product space ${\mathbb{R}}^T$, the cylinder sigma-algebra is generated by coordinate projections. Every event in it depends on at most countably many coordinates.

An event is a measurable set in a probability space: a question with a yes-or-no answer for each outcome. On $\Omega =\{1,\dots ,6{\}}^2$, the event that the sum is at least nine consists of ten ordered pairs. Its probability is therefore $10/36=5/18$ under independent uniform dice. Saying that an event occurred means the actual outcome belongs to that set.

For independent symmetric random variables in a normed vector space and partial sums $S_k$,

Increasing events under a product probability measure are positively associated: $\mathbb{P}(A\cap B)\ge \mathbb{P}(A)\mathbb{P}(B)$.

A concentration inequality bounds the probability that a random variable differs substantially from a typical value such as its expected value or median.

The second moment method uses a random variable's expected value and second moment to establish that a counted or measured quantity is positive with substantial probability. It is useful when a positive average alone could be caused by extremely large values on very rare outcomes. Controlling the variance rules out too much of that concentration in rare outcomes.

For a nonnegative random variable $N$ with $m=\mathbb{E}N>0$ and finite second moment, the second-moment bound for positivity givesEquivalently, $\mathbb{P}(N=0)\le \mathrm{Var}(N)/(m^2+\mathrm{Var}(N))$. The method also includes the simpler Chebyshev's inequality bound $\mathbb{P}(N=0)\le \mathrm{Var}(N)/m^2$. Thus a sequence of such random variables with $\mathrm{Var}(N)/(\mathbb{E}N{)}^2\to 0$ has $N>0$ with probability tending to one. These statements require no independence assumption. The variance argument and its use for random counts are developed in Zhao's lecture notes. ^[ref].

For an original toy count, let $N$ count successes in ten independent Bernoulli trials with success probability $p=1/10$. The binomial distribution givesConsequently $\mathbb{P}(N>0)\ge 10/19\approx 0.526316$. The exact probability is $1-(9/10{)}^{10}\approx 0.651322$. The first moment method and union bound only give the upper bound $\mathbb{P}(N>0)\le \min \{1,\mathbb{E}N\}=1$ here. First and second moments therefore answer complementary existence questions.

For a sum of indicator random variables, the variance of a sum isIndependence makes the covariances vanish, but the formula itself also covers dependent counts. For the same ten success probabilities, let every trial share a single Bernoulli random variable $B$: then $N=10B$, with mean one, variance nine, and $\mathbb{P}(N>0)=1/10$. The second-moment lower bound is exactly $1/10$. Ignoring the positive covariances would produce the false independent-count conclusion. Zhao's discussion of sums of indicators explains why overlapping counted structures require these pairwise terms. ^[ref].

The first moment method uses the expected value of a count to control whether any counted occurrence exists. For a random variable $N$ taking nonnegative integer values, Markov's inequality givesConsequently, $\mathbb{E}N<1$ implies $\mathbb{P}(N=0)\ge 1-\mathbb{E}N>0$, proving that an outcome with no counted occurrence exists. For a sequence of such counts, $\mathbb{E}N\to 0$ shows that $N=0$ with probability tending to one. The first moment method uses information about the average count, while the complementary second moment method also uses its variance.

A lifetime exponential with rate $\mu$ survives for at least duration $u$ with probability $e^{-\mu u}$.

A uniform variable on $[-a,a]$ has mean zero and variance $a^2/3$.