Math 5070 Spring 2006 Homework 2: Sets and Logic Due 1/31/06 Jan Mandel 1. 1/1: If r 6= 0 is rational and x is irrational, prove that r + x and rx are irrational. Solution. Suppose r + x is rational, then x = (r + x) − r is rational because rational numbers form a field, so the difference of two rational numbers is rational. Suppose rx is rational, then x = (rx)/r is rational because rational numbers form a field, so the quotient of two rational numbers is rational. The division is possible because r 6= 0 by assumption. (-1 pt for not using r 6= 0) 2. 1/4: Let E 6= ∅ be a subset of an ordered set. Suppose α is a lower bound of E and β is an upper bound of E. Prove that α ≤ β. Solution. Because α is a lower bound of E, ∀x ∈ E : α ≤ x and because β is an upper bound of E, ∀x ∈ E : β ≥ x. Since E 6= ∅, there exists x ∈ E. Then α ≤ x and x ≤ β so α ≤ β. (It is OK to write α ≤ x ≤ β. -1pt for not using E 6= ∅) 3. Let A be a nonempty set of real numbers which is bounded below. Let −A be the set of all numbers −x, where x ∈ A. Prove that inf A = − sup(−A). Solution. Let a ∈ R. We have ⇐⇒ ⇐⇒ ⇐⇒ ⇐⇒ ⇐⇒ ⇐⇒ a = − sup(−A) −a = sup(−A) (multiplying by -1) (∀x ∈ −A : x ≤ −a) ∧ (∀y < −a∃x ∈ −A : x > y) (from the definition of sup ) (∀ − x ∈ A : x ≤ −a) ∧ (∀y < −a∃ − x ∈ A : x > y) (from the definition of − A) (∀x ∈ A : −x ≤ −a) ∧ (∀ − y < −a∃x ∈ A : −x > −y) (substitute − x for x and − y for y) (∀x ∈ A : x ≥ a) ∧ (∀y > a∃x ∈ A : x < y) (multiplying the all inequalities by − 1) a = inf A (since this is the definition of infimum) Since A is nonempty and bounded below, inf A exists. Hence inf A = − sup(−A). (-1 pt for not using that inf A exists.) 1 (a) The assumption that A is nonempty is not needed for the equivalences above. But all the equivalences prove is that − sup(−A) exists if and only if inf A does, and if they do, then they equal. (b) I have of course derived this proof by writing the definitions of a = − sup(−A) and a = inf A and comparing them; this should be the starting point of any attempt at the solution. (c) Another way to structure this proof is: Since A is nonempty and bounded below, a = inf A exists. Then a = − sup(−A) =⇒ −a = sup(−A) (multiplying by -1) etc., replacing all ⇐⇒ above by =⇒. (d) A more verbal way of writing the same proof: Since A is nonempty and bounded below, a = inf A exists. Then a is lower bound on A and no number y > a is a lower bound on A, etc., . . . hence a = − sup(−A). 4. 1/11 If z ∈ C prove that there exists an r ≥ 0 and w ∈ C with |w| = 1 such that z = rw. Are w and r always uniquely determined by z? Solution. Suppose that z = rw, r ≥ 0, |w| = 1. Then |z| = |rw| = |r| |w| = r, (1) so r is determined uniquely by r = |z|. If z 6= 0, then dividing z = rw by r, we get z |z| r z = = 1. w = , |w| = = r r |r| r So, in conclusion, if z 6= 0, then r = |z| , w = z , r > 0, r (2) and r and w are unique. We need to verify that (2) implies z = rw: rw = |z| z z = r = z. r r If z = 0, then z = rw with r = 0 and any w ∈ C with |w| = 1; so, in this case, w is not unique, but r is still unique by (1). (-1 pt for missing the uniqueness of r) 5. 1/13 If x, y are complex, prove that ||x| − |y|| ≤ |x − y|. Solution. Using the triangle inequality, |x| − |y| ≤ |x − y| 2 because |x| + |x − y| = |x| + |y − x| ≤ |x + (y − x)| = |y|. Switching the notation x and y, |y| − |x| ≤ |y − x| = |x − y| which gives − |x − y| ≤ |x| − |y| hence − |x − y| ≤ |x| − |y| ≤ |x − y| , so ||x| − |y|| ≤ |x − y|. 6. 11/15 Under what conditions does equality hold in Schwarz inequality? Solution. From 1.35: With X X X 2 2 A= |aj | , B = |bj | , C = aj bj , it was proved that X 2 2 |Baj − Cbj | = B AB − |C| . 2 (3) 2 Schwarz inequality is |C| ≤ AB. Suppose that AB = |C| . First, if B = 2 0, then (bj ) = 0. Now consider the case B 6= 0. Then, from (3), AB = |C| P 2 if and only if |Baj − Cbj | = 0, which implies that Baj − Cbj = 0 for all j.Dividing by B 6= 0, C ∀j : aj = bj . B So, we have proved 2 AB = |C| =⇒ (bj ) = 0 ∨ ∃t ∈ C : (aj ) = t (bj ) . (4) Now we need to show the converse implication. If (bj ) = 0, then B = C = 2 0, so AB = |C| = 0. If (aj ) = t (bj ), then X X 2 2 2 2 A = |aj | = |t| |bj | = |t| B X X X 2 C = aj bj = tbj bj = t |bj | = tB so AB 2 |C| 2 2 = |t| BB = |t| B 2 2 2 = |tB| = |t| B 2 2 and AB = |C| . (-1pt by somehow assuming that t is real) Note that by switching aj and bj and noting that (bj ) = 0 can be written as (bj ) = t (aj ) with t = 0, (4) can be equivalently expressed “equality holds if and only if one of (aj ) or (bj ) is a (complex) multiple of the other.” Let’s prove this. (Not required, unless the result is somehow written in this form.) 3