Math Rendering Rules

Verification page for inline and display LaTeX rendering via KaTeX.

Math Rendering Rules

This page exercises inline math, display math, and common LaTeX constructs to verify KaTeX rendering in markdown.

Inline Variables and Fonts

$a$, $x$, $y$, $\mathcal{R}$, $\mathcal{S}$, $\mathbb{N}$, $\mathbb{Z}$, $\mathbb{Q}$, $\mathbb{R}$

Multiplication, Logs, Trig

$2x$, $x \cdot y$, $x \times y$, $\log x$, $\sin x$, $\cos x$

Percentages and Fractions

$x\% = \frac{x}{100} = x/100$

Floor, Ceil, Absolute Value, Factorial

$\lfloor 3.7 \rfloor = 3$, $\left\lceil \frac{x}{2}\right\rceil$, $\lvert -2 \rvert = 2$, $n!$

Sets, Intervals, Binomial

$\{1,2,3\}$, $[0,1]$, $\mathbb{Z}_{\geq 5}$, $\binom{5}{2} = 10$

Display Equations and Environments

$$ 2x + 3y = 7 $$
$$ \left \lceil \frac{x}{2} \right \rceil + \left\lfloor \frac{x}{2} \right\rfloor = x $$

Overlines and Sequences

$\overline{12} = 12$, $\overline{ab} = 12$, $a \neq 0$, $b=2$

Lists with Math

  • $\angle ABC = 60^\circ$ and $\triangle ABC$ use standard geometry notation.
  • $x \in [0,1]$ chosen uniformly at random.
  • Empty set operations: $\sum_{\emptyset} 0 = 0$, $\prod_{\emptyset} 1 = 1$.

Power Towers and Binomials

$2^{3^2} = 512$

$\binom{0}{0} = 1$, $\binom{3}{5} = 0$

Propositional Logic - Quantifiers

Universal: $\forall x$, $\forall x \forall y P(x,y)$
Existential: $\exists x$, $\exists y Q(y)$
Mixed: $\forall x \exists y R(x,y)$

Logical Connectives

$P \land Q$ (and), $P \lor Q$ (or), $P \to Q$ (implies), $\neg P$ (not), $P \leftrightarrow Q$ (iff)

Predicates and Relations

Unary: $P(x)$, $Q(a)$
Binary: $R(x,y)$, $S(a,b)$
Ternary: $T(x,y,z)$

Proof Notation

Entailment: $\Gamma \vdash \phi$ (from $\Gamma$, we derive $\phi$)
Skolem constants: $c, d, a, b$

Herbrand Sets

Base: $\{p(a), p(b), q(a,a), q(a,b)\}$
Model: $\{p(a), q(b,a)\} \subseteq \text{Base}$

Tree Rendering with Mermaid

graph TD
    A["∀x(P(x)→Q(x))"] --> B["P(a)→Q(a)"]
    B --> C["¬P(a)"]
    B --> D["Q(a)"]