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18.100B Problem Set 1
Problems.
1) (10 pts) Prove that there is no rational number whose square is 12.
2) (10 pts) Let S be a nonempty subset of the real
numbers , bounded above. Show that if
u = sup S, then for every natural number n , the number u − 1/n is not an upper
bound of S,
but the number u + 1/n is an upper bound of S.
3) (10 pts) Show that if A and B are bounded subsets of R,
then A∪B is a bounded subset of
R. Show that
supA ∪ B = max{sup A, supB}.
4) (20 pts) Fix b > 1.
a) If m, n, p, q are integers, n > 0, q > 0, and r = m/n = p/q, prove that
Hence it makes sense to define
(How could it have failed to make sense?)
b) Proce that b^{r+s} = b^{r}b^{s} if r, s are rational.
c) If x is real, define B (x) to be the set of all numbers b ^{t}, where t is
rational and t ≤ x.
Prove that
b^{r} = supB (r)
when r is rational . Hence it makes sense to define
b^{x} := supB (x)
for every real x .
d) Prove that b^{x+y} = b^{x}b^{y} for all real x and y.
5) (10 pts) Prove that no order can be defined in the
complex field that turns it into an ordered
field.
(Hint: −1 is a square .)
6) (10 pts) Suppose z = a + bi, w = c + di. Define
z < w if a < c or a = c, b < d
Prove that this turns the set of all complex numbers into
an ordered set. (This is known as a
dictionary order , or lexicographic order.) Does this ordered set have the
least upperbound
property ?
7) (10 pts) Prove that
x + y^{2} + x − y^{2} = 2x^{2} + 2y^{2}
if x ∈ R^{k} and y ∈ R^{k}. Interpret this geometrically, as a statement about parallelograms.
Extra problems:
1) (Another argument showing that
)
Show that, if n^{2} = 2m^{2}, then
(2m − n)^{2} = 2(n − m)^{2}.
Deduce that, if n and m are strictly positive integers
with n^{2} = 2m^{2}, we can find strictly
positive integers n', m' with (n')^{2} = 2(m')^{2} and n' < n. Conclude that the
equation
n^{2} = 2m^{2} has no nonzero integer solutions.
2) Show that the square root of an integer is either an
integer or irrational.
(Hint: Every integer has a unique (up to order) factorization into a product of
prime numbers,
you can use this to show that if n is an integer and a prime p divides n ^{2}, then
p divides n.)
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