Question

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 < y, there exists a number x in R such that x < z < y

The order axioms given are :

-A = (x is an element of R such that -x is an element of A)

-P intersection P = null set (where -P is negative numbers and P is positive numbers)

-P union {0} union P = R

If a and b are elements of P, then a + b is an element of P and ab is an element of P

If b - a is an element of P, then b > a

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 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...

Using the following axioms:
a.) (x+y)+x = x +(y+x) for all x, y in R (associative law of
addition)
b.) x + y = y + x for all x, y elements of R (commutative law of
addition)
c.) There exists an additive identity 0 element of R (x+0 = x
for all x elements of R)
d.) Each x element of R has an additive inverse (an inverse with
respect to addition)
Prove the following theorems:
1.) The additive...

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
⊂...

Prove: Let x,y be in R such that x < y.
There exists a z in R such that x < z <
y.
Given:
Axiom 8.1. For all x,y,z in
R:
(i) x + y = y + x
(ii) (x + y) + z = x + (y + z)
(iii) x*(y + z) = x*y + x*z
(iv) x*y = y*x
(v) (x*y)*z = x*(y*z)
Axiom 8.2. There exists a real number 0 such that
for all...

Use axioms to prove the theorem:
if x and y are non-zero real numbers, then xy does not equal
0

let G be a group of order 18. x, y, and z are elements
of G. if | < x, y >| = 9 and o(z) = order of z = 9, prove
that G = < x, y, z >

Let X, Y ⊂ Z and x, y ∈ Z Let A = (X\{x}) ∪ {x}.
a) Prove or disprove: A ⊆ X
b) Prove or disprove: X ⊆ A 4
c) Prove or disprove: P(X ∪ Y ) ⊆ P(X) ∪ P(Y ) ∪ P(X ∩ Y )
d) Prove or disprove: P(X) ∪ P(Y ) ∪ P(X ∩ Y ) ⊆ P(X ∪ Y )

Let X, Y ⊂ Z and x, y ∈ Z
Let A = (X\{x}) ∪ {x}.
a) Prove or disprove: A ⊆ X
b) Prove or disprove: X ⊆ A
c) Prove or disprove: P(X ∪ Y ) ⊆ P(X) ∪ P(Y ) ∪ P(X ∩ Y )
d) Prove or disprove: P(X) ∪ P(Y ) ∪ P(X ∩ Y ) ⊆ P(X ∪ Y )

A. Let p and r be
real numbers, with p < r. Using the axioms of
the real number system, prove there exists a real number q
so that p < q < r.
B. Let f: R→R be a polynomial
function of even degree and let A={f(x)|x
∈R} be the range of f. Define f
such that it has at least two terms.
1. Using the properties and
definitions of the real number system, and in particular the
definition...

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