SYSTEMATIC MATHEMATICS

Discrete Mathematics

This course contains 40 visible knowledge chapters in eight content-sized units of lengths 4, 4, 4, 5, 5, 5, 9, and 4. How many arrangements are possible without listing them all? Can a network connect every point at least cost? Will a step-by-step rule eventually stop? Discrete mathematics studies separate objects such as choices, strings, schedules, and networks rather than continuously varying lengths. Start by turning ordinary claims into precise yes-or-no statements, truth tables, normal forms, and quantified sentences. Then build sets, functions, relations, proof methods, counting, Pascal’s identity and the binomial theorem, induction, recursion, loop invariants, recurrences, generating functions, asymptotic bounds, undirected and directed reachability, weighted paths, spanning trees, matchings, colorings, finite probability, and expectation in dependency order. Every new symbol is named before it is used, every major claim includes a checkable argument, and wrong initialization, negative weights, disconnected graphs, reversed arrows, greedy traps, odd cycles, and missing initial values are stated as boundaries rather than hidden. The scheduler capstone remains the final chapter after every required directed-graph tool.

Before this course: Algebra & Functions Chapters 1–3: variables, equations, inequalities, and solution sets. No prior proof, programming, probability, or graph course is assumed.

COURSE FACTSLevel, chapters, units, prerequisite, and outcome
Chapter 1

Propositions have truth values in an interpretation

Objective: Why is a false premise not a counterexample to p→q?

A proposition is a declarative statement that is true or false once its terms and context are fixed.

Objects and certificate: A proposition has one truth value in a fixed interpretation; ¬p, p∧q, p∨q, and p→q form new propositions by defined truth rules. The result to establish is: The implication p→q is false exactly when p is true and q is false. Decide whether its evidence is a truth table, witness, bijection, partition, induction chain, invariant, recurrence, graph trace, or normalized probability.

The worked question is: Evaluate (p→q)∧p when p=true and q=false. Begin with “p→q is false in the one violating row.” and complete the reconstruction with “false∧true is false.” Then compare the certificate with this boundary: “Open the door” is a command, not a proposition with a truth value.

The implication p→q is false exactly when p is true and q is false.

An implication promises q for every case in which p holds.

The row p=true,q=false violates that promise.

If p is false there is no active p-case, and if q is true the promised result holds.

Evaluate (p→q)∧p when p=true and q=false.

  1. p→q is false in the one violating row.
  2. p itself is true.
  3. false∧true is false.

Result: The compound proposition is false.