23 lines
827 B
Markdown
23 lines
827 B
Markdown
# ECE 208: Discrete Math 2
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## Hilbert system rules
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**Axioms:**
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- $\vdash (A\implies (B\implies A))$
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- $\vdash (A\implies (B\implies C))\implies ((A\implies B)\implies (A\implies C))$
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- $\vdash (\neg B\implies\neg A)\implies (A\implies B))$
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**Inference (MP):**
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- $\frac{\vdash A, \vdash A\implies B}{\vdash B}$
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**Derived rules:**
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- Deduction: $\frac{U\cup \{A\}\vdash B}{U\vdash A\implies B}$
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- Contrapositive: $\frac{U\vdash \neg B\implies\neg A}{U\vdash A\implies B}$ (and vice versa)
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- Transitivity: $\frac{u\vdash A\implies B, U\vdash B\implies C}{U\vdash A\implies C}$
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- Exchange of antecedent: $\frac{U\vdash A\implies (B\implies C)}{U\vdash B\implies (A\implies C)}$
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- Double negation: $\frac{U\vdash \neg\neg A}{U\vdash A}$ (and vice versa)
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- Reductio ad absurdum: $\frac{U\vdash\neg A\implies false}{U\vdash A}$
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