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Unformatted text preview: LECTURE 6: BASIC PROOFS OF IMPLICATIONS ( § 3.1-3.3) Comedy is simply a funny way of being serious. Peter Ustinov (1921 - 2004) 1. Terminology Some things in math we take for granted. These are called axioms . In this class we will assume quite a few things, such as the truth tables already presented, that sets work correctly, and that numbers obey some basic properties. In other classes, you might assume more or less axioms. When we have big statements we want to verify using the axioms we call them theorems . For simple statements we will just call them results in this class. A corollary is a result that follows directly from a previous result. A lemma is a really simple statement which we prove so we can use it to prove another result. (We think of it as a helping result.) 2. Trivial Proofs We want a method to prove statements of the form ∀ x ∈ S : P ( x ) ⇒ Q ( x ). In some cases this is easier than in other cases. Example 1. Prove the statement: For (all) x ∈ R , if 3 x + 1 > 5 then 1 > 0. Proof. The conclusion is always true. Thus, the implication is true. Usually the conclusion is not always true. But when it is, we say the implication is trivially true. The proof above is called a trivial proof . Here is another example. Result. Let x ∈ R . If x < then x 2 + 1 > . Proof. Since x 2 ≥ 0 for any real number x , then we have x 2 + 1 > x 2 ≥ 0. Hence the conclusion is always true....
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This note was uploaded on 03/08/2011 for the course MATH 334 taught by Professor Smith during the Spring '11 term at Vanderbilt.
- Spring '11