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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 .)

ex + 1

ex 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 number 3 2 is irrational.

2. The number log2 (3) is irrational.

3. Let x R satisfy x7 + 5x2 3 = 0. Then x is irrational.

Problem 3. Prove if a and b are positive real numbers, then a + b 2ab.

Problem 4. Prove the equation 4x2 + 5y2 = 7 has no integer solutions. (Hint: Think about this equation modulo 4.)

Problem 5. Let L be a real number. Recall that a sequence a1 , a2, a3, . . . R is said to converge to L if > 0, N N, n > N, |an L| < .

2an

1. (This part is optional.) Let a1 = 1 and an+1 = an + 1 . Prove that the sequence a1, a2, . . .

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 a1 = 1 and an+1 = f (an ). Prove that the sequence a1, a2 , a3, . . . 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.)

an

3. Let a1 be any real number that is not 0 or 1, and define a sequence by an+1 = 1 1

for every

n 1. Prove that the sequence a1 , a2 , . . . does not converge. You can use the following fact

without proof: Let f be a continuous function, a1 a element in the domain of f , and define the sequence a1, a2, . . . is defined by an+1 = f (an ) 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

2k

2k |n and n

is odd.

3. Consider the function

f : N × N N

(x, y) 2x1(2y 1).

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

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