Question

Prove the following statement: If x ∈ R, then x^{2} + 1
> x.

Answer #1

Prove that R[x]/〈x2 + 1〉 ∼= C as rings.

Using only definition 4.3.1 (continuity), prove that
f(x)=x2+3x+4 is continuous on R.

Prove that the function f : R \ {−1} → R defined by f(x) = (1−x)
/(1+x) is uniformly continuous on (0, ∞) but not uniformly
continuous on (−1, 1).

Prove this statement or show why it's false (provide a counter
example)
∀x(R(x) ∨ S(x)) → (∃xR(x) ∨ ∃yS(y))

Prove the following: Theorem. Let X be a set and {Xi
⊆ X : i ∈ I} be a partition of X. Then R = { (x1,
x2) ∈ X × X : ∃i ∈ I,(x1 ∈ Xi) ∧
(x2 ∈ Xi) } is an equivalence relation on
X.

Consider the set V = (x,y) x,y ∈ R with the following two
operations: • Addition: (x1,y1)+(x2,y2)=(x1 +x2 +1, y1 +y2 +1) •
Scalarmultiplication:a(x,y)=(ax+a−1, ay+a−1). Prove or disprove:
With these operations, V is a vector space over R

let F : R to R be a continuous function
a) prove that the set {x in R:, f(x)>4} is open
b) prove the set {f(x), 1<x<=5} is connected
c) give an example of a function F that {x in r, f(x)>4} is
disconnected

Prove the following: Theorem. Let R ⊆ X × Y and S ⊆ Y × Z be
relations. Then
1. Range(S ◦ R) ⊆ Range(S), and
2. if Domain(S) ⊆ Range(R), then Range(S ◦ R) = Range(S)

Let I= (x2 +2) in Z7 [x] , and let
R be the factor ring Z7 [x] / I.
a) Prove that every element of R can be written in the
form f + I where f is an element of Z7 [x] and
deg(f0< or =2 or f=0. That is,
R={ f + I : f in Z7 [x] and (deg (f) , or=2
or f=0)}

Prove the statement
For all real numbers x, if x − ⌊x⌋ < 1/2 then ⌊2x⌋ =
2⌊x⌋.

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