A constant function is a function whose range consists of

Question

1. Negate the following statement without using any negative words ("no", "not", "none", etc.):

"In every kingdom there is exactly one prince who takes less than three showers a week."

2. Determine whether the given statements are true or false. If the statement is false, give a counterexample.

(a) ∃x ∈ R s.t.     ∀y ∈ R   ∃z ∈ R s.t.            x + y = z.

(b) ∃x ∈ R s.t.     ∀y ∈ R and ∀z ∈ R,          x + y = z.

(c) ∀x ∈ R ∃y ∈ R and ∃z ∈ R s.t.                           z > y ⇒ z > x + y.

3. A constant function is a function whose range consists of exactly one point. Which of the following are valid ways to write the definition of constant function with domain the whole real line? (Hint: There is more than one valid way to write the definition)

Definition. Let f be a function with domain R. We say that f is constant when...

(a) For every x ∈ R there exists a ∈ R such that f(x) = a.

(b) There exists a ∈ R such that for every x ∈ R, f(x) = a.

(c) For every a ∈ R there exists x ∈ R such that f(x) = a.

(d) There exists a ∈ R such that x ∈ R ⇒ f(x) = a.

For any property which is wrong, explain why or give a counterexample.

4. Prove, using mathematical induction, that for every positive integer n,

_i=1Σⁿ(-1)ii² = (-1)ⁿn(n + 1)/2

Note: _i=1Σⁿ(-1)ii² = -1 + 2² - 3² + · · · + (-1)ⁿn²

5. The Fibonacci numbers are defined recursively as follows

f₁ = f₂ = 1 and f_n = f_n-1 + f_n-2 for n ≥ 3.

(The first few Fibonacci numbers are: 1, 1, 2, 3, 5, 8, 13, 21, . . .)

Prove that for every n ≥ 1, the Fibonacci number f_3n is even.

6. Read the proof of the triangular inequality. After you understand this proof, prove the following

√(x² + y²) ≤ |x| + |y|.