Let
be the set of odd integers. Let
.
a) Determine a bijection
from
to
.
b) Is
? Explain.
Let be the set of odd integers. Let . a) Determine a bijection from to ....
3. (8 marks) Let be the set of integers that are not divisible by 3. Prove that is a countable set by finding a bijection between the set and the set of integers , which we know is countable from class. (You need to prove that your function is a bijection.) We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this image
Let n,
and let
n
be a reduced residue. Let r = odd().
Prove that if r = st for positive integers s and t, then
old(t)
= s.
We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this image
Please show all work:
Let
If x is odd then
If x is even then
Prove that
is true and then solve it.
We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this image
Let be a prime and let be the set of rational numbers whose denominator (when written in lowest terms) is not divisible by . i) Show, with the usual operations of addition and multiplication, that is a subring of . ii) Show that is a subring of . iii) Is a field? Explain. iv) What is where is the set of all fractions with denominator a power of We were unable to transcribe this imageWe were unable to transcribe this...
Note: In the following, if is a set and both and are positive integers, then matrices with entries from . The problem below has many applications. If is a linear map from complex vector space to itself, and is an eigenvalue of , then is a simple eigenvalue of if . 1. Suppose is a vector space of dimension over field where you may assume that is either or , and let be a linear map from to . Show...
Let
be an orthonormal set of a Hilbert space. Let
and
be two vectors in H. Show that
converges absolutely, and that
We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this image
Let be a set. Show that the convex hull of , denoted by , is equal to the set We were unable to transcribe this imageWe were unable to transcribe this imagecvx(S) We were unable to transcribe this image cvx(S)
Let E be the solid bounded by the planes , , , , . Set up all six orders of integration for the evaluation of as an iterated integral. We were unable to transcribe this imageWe were unable to transcribe this imagey=0 We were unable to transcribe this imageWe were unable to transcribe this imagef(x, y, 2)d
Let
Which of the following are TRUE? Select ALL that apply. Please
show all your work.
a.
has a local maximum at
whenever
is an even integer
b.
has a saddle point at
whenever
is an even integer
c.
has a saddle point at
whenever
is an odd integer
d.
has a local minimum at
whenever
is an odd integer
fr, y) = sin(x + 7/2) +y? We were unable to transcribe this imageWe were unable to transcribe this imageWe...
Let be iid observations from , is known and is an unknown real number. Let be the parameter of interest. (a) Find the CRLB for the variance of an unbiased estimator for . (b) Find the UMVUE for . (c) Show that is an unbiased estimator for . (d) Show that . We were unable to transcribe this imageσ2 (μ, ) We were unable to transcribe this imageWe were unable to transcribe this imageg(t) = 211 We were unable to...