Math Rendering Rules

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)"]