Strategy of Mathematical Proof

Strategy of Mathematical Proof

1. Direct Method

   • - The most direct and non ambiguous way to prove conditional statement
   • Proposition; conditional statement of form P
     • • The table shows that if P is false, the statement P⇒Q is automatically true. This means that if we are concerned with showing P⇒Q is true, we don’t have to worry about the situations where P is false (as in the last two lines of the table) because the statement P⇒Q will be automatically true in those cases.
     • Outline for Direct Proof
   • Strategy of direct method
     • There is no exact strategy in direct proof. It is the most straightforward method if the given proposition has the form of conditional statement

2. Indirect Method

   • Contrapositive
     • An alternative to direct proof
     • Strategy to prove the statement by contrapositive proof
       • • 3. Example
     • Outline for Contrapositive Proof
   • Contradiction
     • Sometimes false are needed to prove the truth
       • This way requires us to assume that the statement we want to prove is false and then show that this assumption leads to nonsense. We then led to conclude that we were wrong to assume the statement was false so that proves that the statement must be true.
     • Proposition of Contradiction
       • • Based on the truth table above, in order to prove a statement P, it suffices to instead prove the conditional statement (∼P)⇒(C∧∼C). This can be done with direct proof: Assume ∼P and deduce C∧∼C
       • Outline for Contradiction
     • Strategy in this proving
       • Despite the power of proof by contradiction,it’s best to use it only when the direct and contrapositive approaches do not seem to work. The reason for this is that a proof by contradiction can often have hidden in it a simplercontrapositivep roof,and if this is the case it’s better to go with the simpler approach. Consider the following example

3. Mathematical Induction

   • Outline for Proof by Induction
   • Strategy of Mathematical Induction
     • Mathematical induction is designed to answer just the kind of question when we have a set of statements S1, S2, S3,...,Sn,..., and we need to prove that they are all true. Consider the following example

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