Question

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 identity in R is unique.

2.) Every real number x has a unique additive inverse

3.) x+y = x + z if and only if y = z, assuming that x, y, and z are elements R

4.) Suppose x,y are elements of R. Then the following two statements hold

(i) If x+y = x, then y = 0

(ii) If x + y = 0, then y = -x

Answer #1

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 <...

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...

Consider the set Q(√3) ={a+b√3| a,b∈Q}. We have the associative
properties of usual addition and usual multiplication from the
field of real number R.
a)Show that Q (√3) is closed under addition, contains the
additive identity (0,zero) of R, each element contains the additive
inverses, and say if addition is commutative. What does this tell
you about (Q(√3,+)?
b) Prove that Q(√3) is a commutative ring with unity 1
c) Prove that Q(√3) is a field by showing every nonzero...

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...

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,...

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...

Suppose that R is a commutative ring without
zero-divisors.
Let x and y be nonzero elements.
1. Suppose that x has infinite additive order.
Show that y also has infinite additive order.
2. Suppose that the additive order of x is n.
Show that the additive order of y is at most n.
3.Show that all the nonzero elements of R have the same additive
order.

Define a new operation of addition in Z by x ⊕ y = x + y − 1 and
a new multiplication in Z by x y = 1.
• Is Z a commutative ring with respect to these operations?
• Find the unity, if one exists.

1. Consider the set (Z,+,x) of integers with the usual addition
(+) and multiplication (x) operations. Which of the following are
true of this set with those operations? Select all that are true.
Note that the extra "Axioms of Ring" of Definition 5.6 apply to
specific types of Rings, shown in Definition 5.7.
- Z is a ring
- Z is a commutative ring
- Z is a domain
- Z is an integral domain
- Z is a field...

A function f”R n × R m → R is bilinear if for all x, y ∈ R n and
all w, z ∈ R m, and all a ∈ R: • f(x + ay, z) = f(x, z) + af(y, z)
• f(x, w + az) = f(x, w) + af(x, z) (a) Prove that if f is
bilinear, then (0.1) lim (h,k)→(0,0) |f(h, k)| |(h, k)| = 0. (b)
Prove that Df(a, b) · (h, k) = f(a,...

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