SYSTEMATIC MATHEMATICS

Proof, Logic & Set Theory

This course contains 38 visible knowledge chapters in eight content-sized units of lengths 5, 5, 6, 5, 4, 4, 5, and 4. What exactly must be shown when a problem says “prove it”? Begin with short everyday claims such as “every student submitted” and “some student submitted,” and learn why changing one word changes the claim. Symbols are read as complete sentences before they are manipulated. The course separates arbitrary universal objects from fresh existential witnesses, separates syntax from meaning, and states which ZF or ZFC construction rule licenses a new set instead of appealing to an unrestricted universal collection. It then builds cardinality through Cantor diagonalization, Cantor–Schröder–Bernstein, and explicit countable-union encodings, distinguishes finite choice from the Axiom of Choice, checks quotient operations for representative independence, and extends induction through limit stages. Direct, indirect, vacuous, constructive, nonconstructive, recursive, invariant, and transfinite proofs are taught one obligation at a time. Every theorem application must match its object types and every hypothesis, and the bilingual cardinality capstone remains the final chapter.

Before this course: Completed Discrete Mathematics. The course assumes its elementary logic, sets, functions, direct proof, contradiction, counting, induction, recursion, and graph vocabulary, but no calculus, linear algebra, abstract algebra, or real analysis.

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

A statement binds every variable to a universe

Objective: Why would ∃y∀x, x+y=0 say something stronger?

A formula becomes a truth-valued mathematical statement only after its free variables are assigned or quantified and its symbols receive an interpretation.

Objects and obligations: A variable occurrence is bound when it lies inside its quantifier’s scope; otherwise it is free. The result to establish is: For real x, x²≥0 is true, while x²≥0 with x free is an open formula rather than a complete proposition. Separate hypotheses from the target, bind every variable, and state whether the proof needs an arbitrary object, a witness, or a counterexample.

The worked question is: Close the formula x+y=0 so it states that every real x has an additive inverse. Start by “Declare x,y∈ℝ.” and close the argument with “Write ∀x∈ℝ ∃y∈ℝ, x+y=0.” Then compare the resulting proof obligation with this boundary: Changing the universe to complex numbers makes the order symbol ≥ undefined without extra structure.

For real x, x²≥0 is true, while x²≥0 with x free is an open formula rather than a complete proposition.

Declare the universe ℝ. This fixes the active universe, rule, or inference.

Bind x universally: ∀x∈ℝ, x²≥0.

Every real square is a product of equal-sign factors and is nonnegative.

Close the formula x+y=0 so it states that every real x has an additive inverse.

  1. Declare x,y∈ℝ. This verifies the stated construction or implication.
  2. Choose quantifier order ∀x∃y.
  3. Write ∀x∈ℝ ∃y∈ℝ, x+y=0.

Result: The closed statement is ∀x∈ℝ ∃y∈ℝ such that x+y=0.