Question

(a) If r^2 < 2 and s^2 < 2, show that rs < 2.

(b) If a rational t < 2, show that t = rs for some rational r, s with r^2 < 2, s^2 < 2.

(c) Why do (a) and (b) show that that √2 * √2 = 2?

Answer #1

3. Assume √2 ∈R. Let S = { rational numbers q : q < √2 }.
(a)(i) Show that S is nonempty. (ii) Prove that S is bounded from
above, but is not bounded from below. (b) Prove that supS = √2.

f(k) = f(i) f(j) = (rs)(r^2) = ____
f(-k) = f(j) f(i) = (r^2)(rs) = ____
Are the two the same?

Let T be a linear transformation from Rr to
Rs .
Determine whether or not T is one-to-one in each of the following
situations:
1. r > s
2. r < s
3. r = s
A. T is not a one-to-one transformation
B. T is a one-to-one transformation
C. There is not enough information to tell
Explain reason clearly plz

3. (a) Consider R 3 over R. Show that the vectors (1,
2, 3) and (3, 2, 1) are linearly independent. Explain why they do
not form a basis for R 3 .
(b) Consider R 2 over R. Show that the vectors (1, 2),
(1, 3) and (1, 4) span R 2 . Explain why they do not form a basis
for R 2 .

Show that for a subset S
of R, bd(S) cannot contain an interval
((a,b),[a,b),(a,b],[a,b]).

2. Let A = {p, q, r, s}, B = {k, l, m, n}, and C = {u, v, w},
Define f : A→B by f(p) = m, f(q) = k, f(r) = l, and f(s) = n, and
define g : B→C by g(k) = v, g(l) = w, g(m) = u, and g(n) = w. Also
define h : A→C by h = g ◦ f. (a) Write out the values of h. (b) Why
is it that...

Let S and T be nonempty subsets of R with the following
property: s ≤ t for all s ∈ S and t ∈ T.
(a) Show that S is bounded above and T is bounded below.
(b) Prove supS ≤ inf T .
(c) Given an example of such sets S and T where S ∩ T is
nonempty.
(d) Give an example of sets S and T where supS = infT and S ∩T
is the empty set....

Given the following information about a stock fund S and a bond
fund B
E(rS) = 15%,
sS
= 35%,
E(rB) = 9%,
sB
= 23%,
r
= 0.15,
Rf
= 5.5%
a. Calculate the risk (s) of a portfolio made of 60% invested in
the stock fund and 40% in the bond fund call this new portfolio
L
b. Calculate the weights of the minimum variance portfolio. Is
there any benefit from diversification? Explain.
c. If the correlation coefficient...

Answer b please...
Let R be a ring and let Z(R) := {z ∈ R : zr = rz for all r ∈
R}.
(a) Show that Z(R) ≤ R. It is called the centre of R.
(b) Let R be the quaternions H = {a+bi+cj+dk : a,b,c,d ∈ R} and
let S = {a + bi ∈ H}. Show that S is a commutative subring of H,
but there are elements in H that do not commute with elements...

2. Define a relation R on pairs of real numbers as follows: (a,
b)R(c, d) iff either a < c or both a = c and b ≤ d. Is R a
partial order? Why or why not? If R is a partial order, draw a
diagram of some of its elements.
3. Define a relation R on integers as follows: mRn iff m + n is
even. Is R a partial order? Why or why not? If R is...

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