CMPSCI 250 Discussion #2: Translating Quantifiers

19 September 2007

1. (Translate to English, where f is a variable of type "function from the reals to the reals", F is a variable of type "closed, bounded, and nonempty set of reals", all other variables are of type `real`, and DC(f,F) means "f is defined and continuous on F":)

1. DC(f,F) → ∃x:∃y:∀z:((f(x) ≤ f(z)) ∧ (f(y) ≤ f(z)))

If f is defined and continuous on F, there are two real numbers such that the value of f on one is at least as great as all other values of f, and such that the value of f on the other is at least as small as all other values of f. (That is, if f is defined and continuous on F, it achieves a maximum and a minimum value.)

2. (Translate to symbols, using arithmetic on natural numbers and the predicate P(x) meaning "x is prime" -- all variables should be of type `natural`. A perfect square is a natural number that is equal to some natural number multiplied by itself:)

"If a prime number is four times some natural, plus one, then it is the sum of two perfect squares."

∀x:[P(x)∧(∃y:x = 4y + 1)] → ∃y:∃z: x = y⋅y + z⋅z

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 any 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 `natural`. The symbol "⋅" denotes multiplication:)

∀a:∃b: (a > 1) → [(∃c: a = c &sdot b) ∧ ¬[∃d:∃e: (d > 1) ∧ (e > 1) ∧ (e ⋅ d = b)]]

For any natural a there exists a natural b such that if a is greater than 1, there is a number c such that c times b is a and there are not two numbers d and e such that d and e are each greater than one and e times d is b.

More briefly, using terminology from Chapter 3, "Any natural greater than one has a prime divisor".