eifueo/docs/2b/ece208.md

# ECE 208: Discrete Math 2

## Hilbert system rules

**Axioms:**

- $\vdash (A\implies (B\implies A))$
- $\vdash (A\implies (B\implies C))\implies ((A\implies B)\implies (A\implies C))$
- $\vdash (\neg B\implies\neg A)\implies (A\implies B))$

**Inference (MP):**

- $\frac{\vdash A, \vdash A\implies B}{\vdash B}$

**Derived rules:**

- Deduction: $\frac{U\cup \{A\}\vdash B}{U\vdash A\implies B}$
- Contrapositive: $\frac{U\vdash \neg B\implies\neg A}{U\vdash A\implies B}$ (and vice versa)
- Transitivity: $\frac{u\vdash A\implies B, U\vdash B\implies C}{U\vdash A\implies C}$
- Exchange of antecedent: $\frac{U\vdash A\implies (B\implies C)}{U\vdash B\implies (A\implies C)}$
- Double negation: $\frac{U\vdash \neg\neg A}{U\vdash A}$ (and vice versa)
- Reductio ad absurdum: $\frac{U\vdash\neg A\implies false}{U\vdash A}$