real, and DC(f,F)
means "f is defined and continuous on F":)
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.)
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
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.
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".
Last modified 21 September 2007