CMPSCI 250 Discussion #2: Translating Quantifiers

David Mix Barrington

13/15 Febraury 2006

  1. (Translate to English -- all variables are of type natural. The symbol ⋅ denotes multiplication.)

    1. ∀ a: ∃b: [(b>a)∧¬[∃c:∃d:(c>1)∧ (d>1)∧(c⋅d=b)]]

      For every natural a, there exists a natural b such that b is greater than a and there do not exist naturals c and d, each greater than one, such that c times d is b. (Equivalently, "There is a prime number larger than any natural", but we don't have that definition yet.)

    2. (Translate to symbols, using the following predicates (all variables are real numbers): C(a) means "a continually increases", R(a,b) means "a remains less than b", L(a,b) means "a approaches a limit b".)

      If x continually increases but remains less than some number c, it approaches a limit, and this limit is either c or some lesser number. (Hint: Assign a variable to "the limit".)

      C(a)∧R(x,c)→[∃y:L(x,y)∧(y=c ∨ y < c)]

      It is not certain from the English text whether it refers to a particular x and c, in which case they should be free variables as above, or to any x and c, in which case we should have "∀x:∀c:" in front. I actually prefer the second reading with the universal quantifiers.

    3. (Translate to symbols, using "|a-b|<c" to represent "a is within c of b". If you like you may declare some variables to be of type real and some of type positive real.)

      "For every positive real number ε there exists a positive real number δ such that whenever a real number x is within δ of x0, f(x) is within ε of c." What are the free variables in this statement? (Hint: Look carefully at the word "whenever".)

      ∀ε:∃δ:∀x:[(|x-x0|<δ) →(|f(x)-c|<ε)] The free variables are x0 and c, plus f if you consider it to be a variable. The variable x should not be free because "whenever x" means that every possible x should be considered.

    4. (Translate to English, where all variables are of type "node", EP means "the graph has an Euler path", E(a,b) means "there is an edge from a to b", P(a,b) means "there is a path from a to b", and O(a) means "a has an odd number of neighbors"):

      [∀x:∃y:E(x,y)]→[EP↔(∀x:∃y:P(x,y)∧ [∃x:∃y:∀z:(x≠y)∧(O(z)↔((z=x)∨(z=y)))])]

      If for every node x there is a node such that there is an edge from x to y, then the graph has an Euler Path if and only if the following is true: Any two nodes have a path from one to the other, and there are exactly two nodes with an odd number of neighbors.

    Last modified 15 February 2006