Discrete Math: LaTeX Fitness Test

A focused demo showing that PathMX renders the kinds of mathematical expressions used in a typical Discrete Mathematics course.

Every expression below is authored as plain markdown with inline ( $...$ ) or display ($$...$$) LaTeX and rendered by the built-in math plugin. View source on any of these files to see the raw authoring.

Companion files:

Logic Problems -- example problem set (*.problems.md)
Induction Proof -- worked proof (*.proof.md)

Propositional Logic

The standard connectives render as you would expect: $p \land q$ , $p \lor q$ , $\neg p$ , $p \Rightarrow q$ , $p \Leftrightarrow q$ .

De Morgan's laws:

\neg (p \land q) \equiv \neg p \lor \neg q \neg (p \lor q) \equiv \neg p \land \neg q

Contrapositive equivalence:

(p \Rightarrow q) \equiv (\neg q \Rightarrow \neg p)

Predicate Logic & Quantifiers

Universal and existential quantifiers over common domains:

\forall x \in ℕ, \exists y \in ℤ such that x + y = 0

Negation of quantified statements:

\neg (\forall x \in S, P (x)) \equiv \exists x \in S, \neg P (x)

A statement about primes:

\forall n \in ℕ, n > 1 \Rightarrow \exists p \in ℙ, p | n

Sets

Basic set operations with typical notation:

Union: $A \cup B = {x : x \in A \lor x \in B}$
Intersection: $A \cap B = {x : x \in A \land x \in B}$
Difference: $A ∖ B = {x : x \in A \land x \notin B}$
Complement: $A = U ∖ A$
Empty set: $\emptyset$
Cardinality: $| A |$
Power set: $𝒫 (A) = {S : S \subseteq A}$
Cartesian product: $A \times B = {(a, b) : a \in A, b \in B}$

The distinguished number sets: $ℕ, ℤ, ℚ, ℝ, ℂ$ .

Inclusion-exclusion for two sets:

| A \cup B | = | A | + | B | - | A \cap B |

Relations & Functions

Function signatures: $f : A \to B$ , $f^{- 1} : B \to A$ .

A relation $R$ on $A$ is an equivalence relation iff it is reflexive, symmetric, and transitive:

\begin{matrix} (reflexive) & \forall a \in A, a R a \\ (symmetric) & \forall a, b \in A, a R b \Rightarrow b R a \\ (transitive) & \forall a, b, c \in A, (a R b \land b R c) \Rightarrow a R c \end{matrix}

Modular arithmetic notation: $a \equiv b (\mod n)$ iff $n | (a - b)$ .

Summations, Products & Sequences

Common closed forms:

\sum_{i = 1}^{n} i = \frac{n (n + 1)}{2} \sum_{i = 1}^{n} i^{2} = \frac{n (n + 1) (2 n + 1)}{6}

Geometric series for $r \neq 1$ :

\sum_{k = 0}^{n - 1} r^{k} = \frac{1 - r^{n}}{1 - r}

Factorial and product notation:

n! = \prod_{k = 1}^{n} k

Combinatorics

Binomial coefficient and its closed form:

(\binom{n}{k}) = \frac{n!}{k! (n - k)!}

Pascal's identity:

(\binom{n}{k}) = (\binom{n - 1}{k - 1}) + (\binom{n - 1}{k})

Binomial theorem:

(x + y)^{n} = \sum_{k = 0}^{n} (\binom{n}{k}) x^{n - k} y^{k}

Permutations of $k$ from $n$ :

P (n, k) = \frac{n!}{(n - k)!}

Aligned Derivations

Proof-style aligned equations, useful for showing step-by-step algebraic manipulation:

\begin{matrix} \sum_{i = 1}^{k + 1} i & = (\sum_{i = 1}^{k} i) + (k + 1) \\ = \frac{k (k + 1)}{2} + (k + 1) \\ = \frac{k (k + 1) + 2 (k + 1)}{2} \\ = \frac{(k + 1) (k + 2)}{2} \end{matrix}

Inline Math in Prose

Authors can weave math into running text naturally. For example: given a finite set $S$ with $| S | = n$ , the power set $𝒫 (S)$ has cardinality $| 𝒫 (S) | = 2^{n}$ , and the number of $k$ -element subsets of $S$ is $(\binom{n}{k})$ . The handshake lemma states that in any graph $G = (V, E)$ , the sum of vertex degrees satisfies $\sum_{v \in V} \deg (v) = 2 | E |$ .

What This Confirms

This page demonstrates that PathMX's markdown authoring pipeline supports, out of the box, the LaTeX patterns an instructor would reach for when authoring Discrete Math content: propositional and predicate logic, set theory, relations, combinatorics, summations, aligned multi-step derivations, and inline math in prose.

Because every file in PathMX is plain markdown, these same expressions remain readable in a text editor, on GitHub, and to any LLM-based authoring assistant -- no separate LaTeX toolchain required.