CS1502 Formal Methods in Computer Science

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CS1502 Formal Methods in Computer Science

• Lecture Notes 9
• Proofs Involving Conditionals

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Methods of Proof

• Want to prove P ? Q.
• Direct Method assume P deduce Q
• Indirect Method assume ?Q deduce ?P
• Proof by contradiction Assume P and ?Q deduce a
  contradiction
• Proof by Induction more about this later

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Informal Conditional Proof

• Prove Tet(a) ? Tet(c) follows from Tet(a) ?
  Tet(b) and Tet(b) ? Tet(c).
• Assume Tet(a) is true. Applying modus ponens to
  the first premise gives us Tet(b). Using modus
  ponens again, this time with the second premise,
  gives us Tet(c). So, we have established Tet(c)
  from our assumption of Tet(a). Thus, Tet(a) ?
  Tet(c)

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In Fitch
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Informal Indirect Proof

• Prove Even(nn) ? Even(n).
• Proving the contrapositive is easier
• Even(n) ? Even(nn)
• Assume Even(n), i.e., Odd(n). Then we can
  express n as 2m 1 for some m. But we see that
  nn 2(2mm 2m) 1, showing that nn is odd.
  Thus, we have shown Even(n) ? Even(nn)

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? Elimination

• P ? Q P Q

? Elim
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? Introduction

• P Q P ? Q

? Intro
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Fitch Festival
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Fitch Festival
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Taut Con resolution step
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Full Proof
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? Introduction

• P Q Q
  P P ? Q

? Intro
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? Elimination

• P ? Q P Q

? Elim