11 - LECTURE 11 EXISTENCE PROOFS AND EXTRA STUFF 5.3-5.5...

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Unformatted text preview: LECTURE 11: EXISTENCE PROOFS AND EXTRA STUFF ( § 5.3-5.5) Forming characters! Whose? Our own or others? Both. And in that momentous fact lies the peril and responsibility of our existence. Elihu Burritt 1. Quick Review Suppose we are trying to prove a statement ∀ x ∈ S, P ( x ) ⇒ Q ( x ). Consider the following 10 options (taken from page 117). 2. Existence Proofs If asked to prove a statement ∃ x ∈ S, P ( x ), we merely have to find a specific x where P ( x ) is true. Sometimes that is as simple as being clever. Sometimes, there are multiple cases to consider. Consider the following examples: Result. There exists an integer which is its own square. Proof. The integer 1 is an example. (So is the integer 0.) Result. There exist irrational numbers a and b so that a b is rational. Before we try to prove this, let’s experiment with some choices. What irrational numbers do you know about? Let’s try a = b = √ 2. Is √ 2 √ 2 irrational? If it is rational, we are done. If it is irrational,irrational?...
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11 - LECTURE 11 EXISTENCE PROOFS AND EXTRA STUFF 5.3-5.5...

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