This** preview**
has intentionally

**sections.**

*blurred***to view the full version.**

*Sign up*
This** preview**
has intentionally

**sections.**

*blurred***to view the full version.**

*Sign up*
This** preview**
has intentionally

**sections.**

*blurred***to view the full version.**

*Sign up*
**Unformatted text preview: **x such that p ( x ) is true. Example: Prove that the following is not valid ﬁrst-order formula: ∃ x ( φ-→ ψ )-→ (( ∃ xφ )-→ ( ∃ ψ )) . January 18 Proofs, Page 4 5.1 Proof methods Proof by induction: to prove “For any integer n ≥ 0, P ( n ).” • Prove that P (0) is true. • Prove that for any integer i ≥ 0, P ( i )-→ P ( i + 1). • Conclude that P ( n ) is true for all n ≥ 0. Correctness: by axiom of natural numbers. Example: Prove that for any n ≥ 0, 0 + 1 + 2 + ··· + n = n ( n + 1) / 2. Prove that 2 b ≥ n 2 for all n ≥ 4. January 18 Proofs, Page 5 5.2 Proofs that go wrong . Mistaken : • To prove p : we know p-→ q and q , therefore, p . • To prove ∀ xP ( x ): here are examples of x that P ( x ) is true, thus P ( x ) must be true for all x . January 18 Proofs, Page 6...

View
Full Document

- Winter '07
- YaoyunShi
- Mathematical proof, Proof methods Proof