WS-9-14-2010 - WS Problem 1. Use De Morgan’s Formulas for...

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Unformatted text preview: 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 p0 be a point on C , and let α be a positive irrational number. Starting from p0 , let us repeatedly move (in a clockwise direction) a distance of α along the circle to obtain the points p1 , p2 , . . .. Prove that the set {pk : k = 0, 1, 2, . . .} is everywhere dense on C . (A set S of points on C is everywhere dense if every arc of positive length on C contains at least one point of S .) [Hint: Show first that the sequence pk is “dense near p0 .”] And finally a much harder (optional) problem about the notion of “everywhere dense.” A function f : R → R is additive if it satisfies the Cauchy equation: f (x + y ) = f (x) + f (y ) for all x, y ∈ R. Prove that the graph of an additive function is either a straight line through the origin or it’s everywhere dense in R2 . (A set S of points in R2 is everywhere dense in R2 if every disk on the plane contains at least one point of S .) ...
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