Question

Suppose F is a field. Use the field axioms to show the following:

(a) For all a,b in F, there exists some c in F such that a+c=b

(b) For all a,b in F where a doesn't equal 0, there exists some c in F such that ac=b

Answer #1

Using field and order axioms prove the following theorems:
(i) 0 is neither in P nor in - P
(ii) -(-A) = A (where A is a set, as defined in the axioms.
(iii) Suppose a and b are elements of R. Then a<=b if and
only if a<b or a=b
(iv) Let x and y be elements of R. Then either x <= y or y
<= x (or both).
The order axioms given are :
-A = (x...

Using field and order axioms prove the following theorems:
(i) Let x, y, and z be elements of R, the
a. If 0 < x, and y < z, then xy < xz
b. If x < 0 and y < z, then xz < xy
(ii) If x, y are elements of R and 0 < x < y, then 0 <
y ^ -1 < x ^ -1
(iii) If x,y are elements of R and x <...

suppose f is a differentiable function on interval (a,b) with f'(x)
not equal to 1. show that there exists at most one point c in the
interval (a,b) such that f(c)=c

Let f : [a,b] → R be a continuous function such that f(x)
doesn't equal 0 for every x ∈ [a,b].
1) Show that either f(x) > 0 for every x ∈ [a,b] or f(x) <
0 for every x ∈ [a,b].
2) Assume that f(x) > 0 for every x ∈ [a,b] and prove that
there exists ε > 0 such that f(x) ≥ ε for all x ∈ [a,b].

5. Suppose that the incenter I of ABC is on the triangle’s Euler
line. Show that the triangle is isosceles.
6. Suppose that three circles of equal radius pass through a
common point P, and denote by A, B, and C the three other points
where some two of these circles cross. Show that the unique circle
passing through A, B, and C has the same radius as the original
three circles.
7. Suppose A, B, and C are distinct...

Real Analysis I
Prove the following exercises (show all your work)-
Exercise 1.1.1: Prove part (iii) of Proposition
1.1.8. That is, let F be an ordered field and x, y,z ∈ F. Prove If
x < 0 and y < z, then xy > xz.
Let F be an ordered field and x, y,z,w ∈ F. Then:
If x < 0 and y < z, then xy > xz.
Exercise 1.1.5: Let S be an ordered set. Let A
⊂...

Let a > 0 and f be continuous on [-a, a]. Suppose that f'(x)
exists and f'(x)<= 1 for all x2 ㅌ (-a, a). If f(a) = a and f(-a)
=-a. Show that f(0) = 0.
Hint: Consider the two cases f(0) < 0 and f(0) > 0. Use
mean value theorem to prove that these are impossible cases.

Prove the following using Field Axioms of Real
Numbers. prove (b^(−1))^−1=b

Use models to show that each of the incidence axioms is
independent of the other three.
Axiom 1: There exists at least 3 distinct noncollinear
points
Axiom 2: Given any two distinct points, there is at least one
line that contains both of them
Axiom 3: Given any two distinct points, there is at most one
line that contains both of them
Axiom 4: Given any line, there are at least two distinct points
that lie on it.

Using field axioms and order axioms prove the following
theorems
(i) The sets R (real numbers), P (positive numbers) and [1,
infinity) are all inductive
(ii) N (set of natural numbers) is inductive. In particular, 1
is a natural number
(iii) If n is a natural number, then n >= 1
(iv) (The induction principle). If M is a subset of N (set of
natural numbers) then M = N
The following definitions are given:
A subset S of R...

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