Review 2 - Elementary Proof Techniques These methods are...

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Elementary Proof Techniques These methods are used when trying to prove that the conditional P = ) Q is true. Vacuous proof: prove that P = ) Q is true by showing that P is always false. This makes the conditional always true, regardless of the truth value of the conclusion Q . Recall that that the truth value of a conditional is not the same as showing that the conclusion is true. Trivial proof: prove that P = ) Q is true by showing that Q is true without using that P is true. Direct Method: prove P = ) Q by starting with the assumptions in P and by following logical deductions, obtain Q Contrapositive Method: uses the fact that P = ) Q is logically equivalent to q Q = ) q P . Sometimes, if Q is a negation itself, it is easier to start by assuming q Q is true. Method by contradiction: Assumes that P and q Q are true, obtain a contradiction. Or, if the statement is "Show P is true", we can assume q P is true and reach a contradiction. Remark 1 as follows: "For all n 2 N , P ( n ) is true" can be written as n 2 N = ) P ( n ) : If P ( n ) exactly for which n P ( n ) is true. For example: n = 1 ; 2 ; 3 = ) P ( n ) . 1
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CHAPTER 3: INDUCTION Used to prove statements involving all or most of the natural numbers. Whenever there are too many natural numbers to verify the statement case-by-case. Principle of Induction: Let P ( n ) be a mathematical statement de- pending on the natural number n . Want to show that P ( n ) is true for all n 2 N (or for all n ± n 0 ). a) verify P (1) is true (or P ( n 0 ) is true) (basis step) b) show that P ( n ) = ) P ( n + 1) for (arbitrary) n 2 N (or for n ± n 0 ) (inductive step) ; here P ( n ) is the induction hypothesis . Then, by the principle of induction, the statement holds true
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Review 2 - Elementary Proof Techniques These methods are...

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