Question

Let f be a function which maps from the quaternion group, Q, to itself by f (x) = i ∙x, for i∈ Q and each element x in Q. Show all work and explain! (i) Is ? a homomorphism? (ii) Is ? a 1-1 function? (iii) Does ? map onto Q?

Answer #1

Let A be a finite set and let f be a surjection from A to
itself. Show that f is an injection.
Use Theorem 1, 2 and corollary 1.
Theorem 1 : Let B be a finite set and let f be a function on B.
Then f has a right inverse. In other words, there is a function g:
A->B, where A=f[B], such that for each x in A, we have f(g(x)) =
x.
Theorem 2: A right inverse...

Let G be a group and let p be a prime number such that
pg = 0 for every element g ∈ G.
a. If
G is commutative under multiplication, show that the mapping
f : G → G
f(x) =
xp
is a homomorphism
b. If G is
an Abelian group under addition, show that the mapping
f : G → G
f(x) = xpis a homomorphism.

Let G be a cyclic group, and H be any group. (i) Prove that any
homomorphism ϕ : G → H is uniquely determined by where it maps a
generator of G. In other words, if G = <x> and h ∈ H, then
there is at most one homomorphism ϕ : G → H such that ϕ(x) = h.
(ii) Why is there ‘at most one’? Give an example where no such
homomorphism can exist.

let g be a group. Call an isomorphism from G to itself a self
similarity of G.
a) show that for any g ∈ G, the map cg :
G-> G defined by cg(x)=g^(-1)xg is a self similarity
group
b) If G is cyclic with generator a, and sigma is a self
similarity of G, prove that sigma(a) is a generator of G
c) How many self-similarities does Z have? How many self
similarities does Zn have? How many self-similarities...

Let Let A = {a, e, g} and B = {c, d, e, f, g}. Let f : A → B and
g : B → A be defined as follows: f = {(a, c), (e, e), (g, d)} g =
{(c, a), (d, e), (e, e), (f, a), (g, g)}
(a) Consider the composed function g ◦ f.
(i) What is the domain of g ◦ f? What is its codomain?
(ii) Find the function g ◦ f. (Find...

a.)Consider the function f (x) = 3x/ x^2 +1
i) Evaluate f (x+1), and f (x)+1. Explain the difference. Do the
same for f (2x) and 2f (x).
ii) Sketch y = f (x) on the interval [−2, 2].
iii) Solve the equations f (x) = 1.2 and f (x) = 2. In each
case, if a solution does not exist, explain.
iv) What is the domain of f (x)?
b.)Let f (x) = √x −1 and g (x) =...

Let X and Y be metric spaces. Let f be a continuous function
from X onto Y, that is the image of f is equal to Y. Show that if X
is compact, then Y is compact

4. (30) Let C be the ring of complex numbers,and letf:C→C be the
map defined by
f(z) = z^3.
(i) Prove that f is not a homomorphism of rings, by finding an
explicit counterex-
ample.
(ii) Prove that f is not injective.
(iii) Prove that the principal ideal I = 〈x^2 + x + 1〉 is not a
prime ideal of C[x].
(iv) Determine whether or not the ring C[x]/I is a field.

Let f : R → R be defined by f(x) = x^3 + 3x, for all x. (i)
Prove that if y > 0, then there is a solution x to the equation
f(x) = y, for some x > 0. Conclude that f(R) = R. (ii) Prove
that the function f : R → R is strictly monotone. (iii) By
(i)–(ii), denote the inverse function (f ^−1)' : R → R. Explain why
the derivative of the inverse function,...

Let f be a function for which the first
derivative is f ' (x) = 2x 2 - 5 / x2 for x
> 0, f(1) = 7 and f(5) = 11. Show work for all
question.
a). Show that f satisfies the hypotheses of the Mean
Value Theorem on [1, 5]
b)Find the value(s) of c on (1, 5) that satisfyies the
conclusion of the Mean Value Theorem.

ADVERTISEMENT

Get Answers For Free

Most questions answered within 1 hours.

ADVERTISEMENT

asked 17 minutes ago

asked 29 minutes ago

asked 53 minutes ago

asked 1 hour ago

asked 2 hours ago

asked 2 hours ago

asked 3 hours ago

asked 3 hours ago

asked 4 hours ago

asked 4 hours ago

asked 4 hours ago

asked 4 hours ago