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## Description

Problem 1. Let f : A → B and g : B → A be functions.

1. Prove if f ◦ g is injective then g is injective.

2. Prove if f ◦ g is surjective then f is surjective.

3. Show that the function f : (0, +∞) → (0, +∞) defined by f (x) = ln

(Hint: Calculate f ◦ f .)

e^{x} + 1

e^{x} − 1

is bijective.

x

4. Let f : R − {0, 1} → R − {0, 1} be the function defined by f (x) = 1 − ^{1} . Prove that f (x) is

bijective. (Hint: Calculate f ◦ f ◦ f .)

Problem 2. Prove each of the following.

_{1. } _{The} _{n}_{u}_{m}_{b}_{er } ^{√}^{3} _{2} _{is} _{irrational.}

2. The number log_{2} (3) is irrational.

3. Let x ∈ R satisfy x^{7} + 5x^{2} − 3 = 0. Then x is irrational.

_{Problem} _{3. } _{Pr}_{ov}_{e } _{if} _{a} _{and} _{b} _{are} _{p}_{ositi}_{v}_{e} _{real} _{n}_{u}_{m}_{b}_{ers, } _{then } _{a} _{+} _{b} _{≥} _{2}^{√}_{ab}_{.}

Problem 4. Prove the equation 4x^{2} + 5y^{2 } = 7 has no integer solutions. (Hint: Think about this equation modulo 4.)

Problem 5. Let L be a real number. Recall that a sequence a_{1} , a_{2}, a_{3}, . . . ∈ R is said to converge to L if ∀ > 0, ∃N ∈ N, ∀n > N, |a_{n} − L| < .

2^{a}^{n}

1. (This part is optional.) Let a_{1 } = 1 and a_{n+1} = a_{n} + __ ____ __^{1 }__ __ . Prove that the sequence a_{1}, a_{2}, . . .

does not converge.

2. Recall that a real-valued function f is called continuous at x = c if

∀ > 0, ∃δ > 0, (|x − c| < δ =⇒ |f (x) − f (c)| < ).

Let f : (0, ∞) → (0, ∞) be a continuous function such that f (x) > x for all x. Define a sequence by a_{1 } = 1 and a_{n+1} = f (a_{n} ). Prove that the sequence a_{1}, a_{2} , a_{3}, . . . does not converge. (Hint: Suppose for a contradiction that there is a limit, L. Use continuity to show that if x is sufficiently close to L, then f (x) > L.)

^{a}_{n}

3. Let a_{1 } be any real number that is not 0 or 1, and define a sequence by a_{n+1} = 1 − __ __^{1}

for every

^{n} ^{≥} ^{1. } ^{Pr}^{ov}^{e } ^{that} ^{the } ^{sequence} ^{a}_{1} ^{,} ^{a}_{2} ^{,} ^{.} ^{.} ^{.} ^{d}^{o}^{es} ^{not } ^{co}^{nv}^{erge. } ^{Y}^{ou} ^{can} ^{use} ^{the } ^{foll}^{o}^{wing} ^{fact}

without proof: Let f be a continuous function, a_{1 } a element in the domain of f , and define the sequence a_{1}, a_{2}, . . . is defined by a_{n+1} = f (a_{n} ) for every n ≥ 1. If the sequence converges to a limit L, then f (L) = L. (Hint: your solution to Problem 1, part 4 might help.)

Problem 6. A set A is called countably infinite if there exists a bijection A → N.

1. Let A be a set. Show that A is countably infinite if and only if there exists a bijection N → A.

2. Show by strong induction (see HW7) that for every n ∈ N, there exists k ∈ Z, k ≥ 0 such that

2^{k}

_{2}^{k} _{|}_{n} _{and } ^{n}

is odd.

3. Consider the function

f : N × N −→ N

_{(x,} _{y}_{) } _{−}_{→ } _{2}^{x}^{−}^{1}_{(2}_{y} _{−} _{1)}_{.}

Use the function above to prove that N × N is countably infinite.

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