1. Use cardinality to show that between any two rational numbers
there is an irrational number. Hint: Given rational numbers a <
b, first show that [a,b] is uncountable. Now use a proof by
contradiction.
2. Let X be any set. Show that X and P(X) do not have the same
cardinality. Here P(X) denote the power set of X. Hint: Use a proof
by contradiction. If a bijection:X→P(X)exists, use it to construct
a set Y ∈P(X) for which Y...
Let (G,·) be a finite group, and let S be a set with the same
cardinality as G. Then there is a bijection μ:S→G . We can give a
group structure to S by defining a binary operation *on S, as
follows. For x,y∈ S, define x*y=z where z∈S such that μ(z) =
g_{1}·g_{2}, where μ(x)=g_{1} and μ(y)=g_{2}.
First prove that (S,*) is a group.
Then, what can you say about the bijection μ?
Use cardinality to show that between any two rational numbers
there is an irrational number. Hint: Given rational numbers a <
b, first show that [a, b] is uncountable. Now use a proof by
contradiction
8. The cardinality of S is less than or equal to the cardinality
of T, i.e. |S| ≤ |T| iff there is a one to one function from S to
T. In this problem you’ll show that the ≤ relation is transitive
i.e. |S| ≤ |T| and |T| ≤ |U| implies |S| ≤ |U|.
a. Show that the composition of two one-to-one functions is
one-to-one. This will be a very simple direct proof using the
definition of one-to-one (twice). Assume...
3) Prove that the cardinality of the open unit interval, (0,1),
is equal to the cardinality of the open unit cube:
{(x,y,z) E R^3|0<x<1, 0<y<1,
0<Z<1}.
[Hint: Model your argument on Cantor's proof for the
interval and the open square. Consider the decimal expansion of the
fraction 12/999. It may prove handdy]
What discount rate would cause these two investments to have the same NPV?
Year 0 1 2 3 4
Investment A (46) 25 22 16 8
Investment B (46) 14 18 32 10
(Show your answer as a percentage to 2 decimals, like 8.12% is 8.12)
b. Why can't systematic risk be diversified away? (You must
answer in two lines or less. Complete sentences are not
required.)