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WS-9-14-2010 - WS Problem 1 Use De Morgans Formulas for...

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WS Problem 1. Use De Morgan’s Formulas for quantifiers to formally prove the following De Morgan’s Formulas for restricted quantifiers. Let U (the universe of discourse) be given, and let S U . Then, [( x S ) P ( x )] ( x S )( P ( x )) and [( x S ) P ( x )] ( x S )( P ( x )) Definition. A set S R is everywhere dense (dense, for short) if every nonempty open interval contains an element of S : ( a, b R ) [ if a < b, then ( x S ) a < x < b ] ( * ) Theorem (known from calculus). Q , the set of rationals, is everywhere dense. Lemma 1 (known from calculus). 2 is irrational. Lemma 2 (easy). If q is rational, then q 2 is irrational. WS Problem 2. Prove Lemma 2. WS Problem 3. Prove that I , the set of irrationals is everywhere dense. Here is a harder (optional) problem about the notion of “everywhere dense.” Let C be a circle of unit circumference, let p 0 be a point on C , and let α be a positive irrational number. Starting from p 0 , let us repeatedly move (in a clockwise direction) a distance of
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