STRATEGY OF MATHEMATICAL PROOF

STRATEGY OF MATHEMATICAL PROOF

1. DIRECT PROOF

   • OUTLINE
     • The first line of the proof is the sentence “Suppose P.” The last line is the sentence “Therefore Q.” Between the first and last line we use logic, definitions and standard math facts to transform the statement P to the statement Q.
   • EXAMPLE
     • Solution
     • Question
   • DEFINITION
     • A simple way to prove theorems or propositions that have the form of conditional statements.
   • When to use?
     • Direct proof is usually used when we want to proof a statement that almost true or obviously true.

2. MATHEMATICAL INDUCTION

   • DEFINITION
     • Used when we have a set of statements S1,S2,S3,...,Sn,..., and we need to prove that they are all true.
     • Mathematical induction is a technique for proving a statement , a theorem, or a formulathat is asserted about every natural number. By "every", or "all," natural numbers, we mean any one that we might possibly name
   • OUTLINE
     • The first step (1) is called the basis step.
     • The second step (2) is called the inductive step.
   • EXAMPLE
     • QUESTION
     • SOLUTION
   • ANALOGY
     • The force that falling domino exerts on the next domino are same as induction principal

3. CONTRAPOSITIVE PROOF

   • DEFINITION
     • An integer x is called even (respectively odd) if there is another integer k for which x = 2k (respectively 2k+1). Two integers are said to have the same parity if they are both odd or both even.
   • EXAMPLE
     • Question and Solution
   • METHOD
     • Assume Not Q and prove Not P
   • OUTLINE
     • p →q is logically equivalent to ~ q → ~p
   • TRUTH TABLE
   • STRATEGY

4. CONTRADICTION PROOF

   • DEFINITION
     • Examine an alternative to direct proof called contrapositive proof. Like direct proof, the technique of contrapositive proof is used to prove conditional statements of the form “If P, then Q.”
   • OUTLINE
     • ~p v q is logically equivalent to p ˄ ~ q p →q ≡ ~p v q ~(p → q) ≡ ~(~p ˅ q) ≡ p˄ ~q
   • TRURH TABLE
   • EXAMPLE
     • Question and Solution
   • METHOD
     • Assume P and not Q and prove some sort of contradiction.
   • STRATEGY
     • The usual proof of the lemma does use contradiction: one assumes that the conclusion is false and applies the mean value theorem.

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