Question

if f belongs to R[a,b] and k belongs to R show that kf belongs to R[a,b]

Answer #1

show that f: R^2->R^2 be f(x,y)= (cosx + cosy, sinx + siny).
show that f is locally invertible near all points (a,b)such that
a-bis not = kpi where k in z and all other points have no local
inverse exists

using the epsilon deal is continous, show that the function
f(x)=x if x belongs to the rational numbers
0 if x belongs to the irrational numbers
is continous at x=0

Let f: [a,b] to R be continuous and strictly increasing on
(a,b). Show that f is strictly increasing on [a,b].

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

Calculate the concentration of all species in a 0.12 M KF
solution.
[K+], [F−], [HF], [OH−], [H3O+]

Theorem: Given a,b belongs to real number R, with a<b, the
intervals (0,1) and (a,b) have the same cardinality.
Proof: Consider h:(0,1)-----> (a,b), given by h(x)= (b-a)
(x)+a. Finish the proof.

Use the pumping lemma to show that {w | w belongs to {a,
b}*,and w is a palindrome of even length.} is not
regular.

Let R = R[x], f ∈ R \ {0}, and I = (f). Show that R/I is an
integral domain if and only if f is an irreducible polynomial.

Determine whether the following sets define vector spaces over
R:
(a) A={x∈R:x=k^2,k∈R}
(b) B={x∈R:x=k^2,k∈Z}
(c) C ={p∈P^2 :p=ax^2,a∈R}
(d) D={z∈C:|z|=1}
(e) E={z∈C:z=a+i,a∈R}
(f) F ={p∈P^2 : d (p)∈R}

Let R = R[x], f ∈ R \ {0}, and I = (f). Show that R/I is a real
vector space of dimension equal to deg(f).

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