lec-2.pdf

# lec-2.pdf - CS70 Lecture 2 Outline Today Proofs 1 By...

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CS70: Lecture 2. Outline. Today: Proofs!!! 1. By Example (or Counterexample). 2. Direct. (Prove P = Q . ) 3. by Contraposition (Prove P = Q ) 4. by Contradiction (Prove P .) 5. by Cases

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Quick Background and Notation. Integers closed under addition.
Quick Background and Notation. Integers closed under addition. a , b Z = a + b Z

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Quick Background and Notation. Integers closed under addition. a , b Z = a + b Z a | b means “a divides b”.
Quick Background and Notation. Integers closed under addition. a , b Z = a + b Z a | b means “a divides b”. 2 | 4?

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Quick Background and Notation. Integers closed under addition. a , b Z = a + b Z a | b means “a divides b”. 2 | 4? Yes!
Quick Background and Notation. Integers closed under addition. a , b Z = a + b Z a | b means “a divides b”. 2 | 4? Yes! 7 | 23?

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Quick Background and Notation. Integers closed under addition. a , b Z = a + b Z a | b means “a divides b”. 2 | 4? Yes! 7 | 23? No!
Quick Background and Notation. Integers closed under addition. a , b Z = a + b Z a | b means “a divides b”. 2 | 4? Yes! 7 | 23? No! 4 | 2?

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Quick Background and Notation. Integers closed under addition. a , b Z = a + b Z a | b means “a divides b”. 2 | 4? Yes! 7 | 23? No! 4 | 2? No!
Quick Background and Notation. Integers closed under addition. a , b Z = a + b Z a | b means “a divides b”. 2 | 4? Yes! 7 | 23? No! 4 | 2? No! Formally: a | b ⇐⇒ ∃ q Z where b = aq .

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Quick Background and Notation. Integers closed under addition. a , b Z = a + b Z a | b means “a divides b”. 2 | 4? Yes! 7 | 23? No! 4 | 2? No! Formally: a | b ⇐⇒ ∃ q Z where b = aq . 3 | 15
Quick Background and Notation. Integers closed under addition. a , b Z = a + b Z a | b means “a divides b”. 2 | 4? Yes! 7 | 23? No! 4 | 2? No! Formally: a | b ⇐⇒ ∃ q Z where b = aq . 3 | 15 since for q = 5,

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Quick Background and Notation. Integers closed under addition. a , b Z = a + b Z a | b means “a divides b”. 2 | 4? Yes! 7 | 23? No! 4 | 2? No! Formally: a | b ⇐⇒ ∃ q Z where b = aq . 3 | 15 since for q = 5, 15 = 3 ( 5 ) .
Quick Background and Notation. Integers closed under addition. a , b Z = a + b Z a | b means “a divides b”. 2 | 4? Yes! 7 | 23? No! 4 | 2? No! Formally: a | b ⇐⇒ ∃ q Z where b = aq . 3 | 15 since for q = 5, 15 = 3 ( 5 ) . A natural number p > 1, is prime if it is divisible only by 1 and itself.

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Direct Proof (Forward Reasoning). Theorem: For any a , b , c Z , if a | b and a | c then a | b - c . Proof: Assume a | b and a | c
Direct Proof (Forward Reasoning). Theorem: For any a , b , c Z , if a | b and a | c then a | b - c . Proof: Assume a | b and a | c b = aq

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Direct Proof (Forward Reasoning). Theorem: For any a , b , c Z , if a | b and a | c then a | b - c . Proof: Assume a | b and a | c b = aq and c = aq 0
Direct Proof (Forward Reasoning). Theorem: For any a , b , c Z , if a | b and a | c then a | b - c . Proof: Assume a | b and a | c b = aq and c = aq 0 where q , q 0 Z

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Direct Proof (Forward Reasoning).
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