Strategy for Mathematical Proof

Strategy for Mathematical Proof

1. CONTRAPOSITIVE

   • Used when: contraposotive proof is easier, simple direct proof would be problematic.
   • To prove proposition "If p, then q."
     • contrapositive form ∼ Q ⇒∼ P
       • assume ∼ Q is true use this to deduce that ∼ P is true

2. DIRECT

   • Proposition : True statement but not as significant Lemma :prove other theorem Collary:immediateconsequence of a theorem or proposition
   • If p, then q
   • 1. Assume that P is true. 2. Use P to show that Q must be true.
   • Accept these facts without justification or proof.
   • Using Cases
   • Definition : Odd number : 2a + 1, a E

3. INDUCTION

   • use recursion to demonstrate an infinite number of facts in a finite amount of space.
   • condition : when a set of statements is given ex. Fibonacci numbers
   • Step 1: Proof S1 is true
     • Step 2 : Proof Sk →Sk+1 is true
   • examples

4. CONTRADICTION

   • used when direct and contrapositive methods do not seem to work.
   • assume that the statement we want to prove is false, and then show that this assumption leads to nonsense.
     • 1. Assume that P is true.
       • 2. Assume that ~Q is true.
         • 3. Use P and ~Q to demonstrate a contradiction.

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