methods of mathematics proof

methods of mathematics proof

1. DIRECT PROOF

   • If P is true, then Q is true.
   • Tautology: If P is a statement in a proof and Q is logically equivalent to P, we can then write down.
   • P → Q
     • P : Facts or knowledge
     • Q: Statement to be proved
   • Steps:
     • Assume that statement P is true.
       • Use Inference and facts to prove that Q is also true.
         • Show that P>Q is true.
   • Example:
   • Outline

2. INDIRECT PROOF

   • CONTRAPOSITIVE
     • Express the statement to be proved in the form:
       • ∀x∈D, if P(x) then Q(x)
         • Rewiter in the contrapositive form:
           • ∀x∈D, if Q(x) is false then P(x) is false.
             • Prove the contrapositive by a direct proof
     • P → Q
       • logically equivalent to ~ Q → ~ P
     • Example:
     • Outline
   • CONTRADICTION
     • Steps:
       • Assume that the statement to be proved is false.
         • Show that this supposition logically leads to a contradiction
           • Show that the statement to be proved is true
     • Example:
     • Outline

3. MATHEMATICAL INDUCTION

   • Steps:
     • Prove the first statement is true.
       • Show that if any one is true, then the next one is true
   • Example:
   • Outline

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