Practice problems on quantified statements

Math 366Autumn 2010Quantified Statement <strong>Practice</strong> Problems1. Let S = {people at OSU}, M = {all movies}, and let V (x, y) mean “person x has seenmovie y”.Rewrite each of the following statements using S, M, V , ∀, ∃, and logical connectives.(a) Everybody has seen a movie.(b) There is a movie that everyone has seen.(c) There is a movie that no one has seen.(d) Nobody has seen every movie.(e) Every movie has been seen by at least one person.(f) Everyone has seen every movie.(g) There is a person who has seen every movie.(h) Everyone has seen at least two different movies.(i) Two different people have seen the same movie.(j) Someone has never seen a movie.(k) Given any two people, there is a movie that they have both seen.(l) Given any two people, at least one of them has seen a movie.(m) Given any two people, there is a movie that neither of them has seen.(n) There exist two movies that have not been seen by the same person.2. Let D = {0, 1, 3, 4}, E = {1, 3, 4}, and F = {1, 2, 3, 4, 5}. Let U = {D, E, F } andV = {0, 1, 2, 3, 4, 5}.Decide whether each of the following statements is true or false.(a) ∃S ∈ U such that ∀x ∈ V, x ∈ S(b) ∀x ∈ V, ∃S ∈ U such that x ∈ S(c) ∀S ∈ U, ∃x ∈ V such that x ∈ S(d) ∃x ∈ V such that ∀S ∈ U, x ∈ S(e) ∀x ∈ D, ∃y ∈ E such that x + y ∈ F(f) ∃y ∈ E such that ∀x ∈ D, x + y ∈ F(g) ∀y ∈ E, ∃x ∈ D such that x + y ∈ F(h) ∃x ∈ D such that ∀y ∈ E, x + y ∈ F

3. Let f be a function on R. To say that f is uniformly continuous means:for each positive real number ε, there is a positive real number δ such that for alla, b ∈ R, if |a − b| < δ, then |f(a) − f(b)| < ε(a) Write the definition of “f is uniformly continuous” using the quantifiers ∀ and ∃.(b) Write the negation of “f is uniformly continuous” without using quantifiers.4. Let a n be a sequence of real numbers. To say that a n converges to L as n goes toinfinity means:given any positive real number ε, there is a positive integer N such that for alln > N, |a n − L| < ε(a) Write the definition of “a n converges to L” using the quantifiers ∀ and ∃.(b) Write the negation of “a n converges to L” without using quantifiers.