Non-Zero Real Numbers under Multiplication form Abelian Group

From ProofWiki
Jump to navigation Jump to search

Theorem

Let $\R_{\ne 0}$ be the set of real numbers without zero:

$\R_{\ne 0} = \R \setminus \set 0$

The structure $\struct {\R_{\ne 0}, \times}$ is an uncountable abelian group.


Proof 1

Taking the group axioms in turn:


Group Axiom $\text G 0$: Closure

From Non-Zero Real Numbers Closed under Multiplication: Proof 2, $\R_{\ne 0}$ is closed under multiplication.

Note that proof 2 needs to be used specifically here, as proof 1 rests on this result.

$\Box$


Group Axiom $\text G 1$: Associativity

Real Multiplication is Associative.

$\Box$


Group Axiom $\text G 2$: Existence of Identity Element: Real Multiplication Identity is One

The identity element of real number multiplication is the real number $1$:

$\exists 1 \in \R: \forall a \in \R_{\ne 0}: a \times 1 = a = 1 \times a$


$\Box$


Group Axiom $\text G 3$: Existence of Inverse Element: Inverse for Real Multiplication

Each element $x$ of the set of non-zero real numbers $\R_{\ne 0}$ has an inverse element $\dfrac 1 x$ under the operation of real number multiplication:

$\forall x \in \R_{\ne 0}: \exists \dfrac 1 x \in \R_{\ne 0}: x \times \dfrac 1 x = 1 = \dfrac 1 x \times x$


$\Box$


$\text C$: Commutativity

Real Multiplication is Commutative.

$\Box$


Infinite

Real Numbers are Uncountably Infinite.

$\blacksquare$


Proof 2

We have Real Numbers under Multiplication form Monoid.

From Inverse for Real Multiplication, the non-zero numbers are exactly the invertible elements of real multiplication.

Thus from Invertible Elements of Monoid form Subgroup of Cancellable Elements, the non-zero real numbers under multiplication form a group.

From:

Real Multiplication is Commutative
Subset Product within Commutative Structure is Commutative

it follows that this group is also Abelian.

$\blacksquare$


Proof 3

From Non-Zero Real Numbers under Multiplication form Group, $\struct {\R_{\ne 0}, \times}$ forms a group.

$\Box$


From Real Multiplication is Commutative it follows that $\struct {\R_{\ne 0}, \times}$ is abelian.

$\Box$


From Real Numbers are Uncountably Infinite it follows that $\struct {\R_{\ne 0}, \times}$ is an uncountable abelian group.

$\blacksquare$


Sources


This page may be the result of a refactoring operation.
As such, the following source works, along with any process flow, will need to be reviewed.
When this has been completed, the citation of that source work (if it is appropriate that it stay on this page) is to be placed above this message, into the usual chronological ordering.
In particular: Determine whether or not the actual proof is given here, or just a statement of the result. If the former, move this citation onto that proof page.
If you have access to any of these works, then you are invited to review this list, and make any necessary corrections.
To discuss this page in more detail, feel free to use the talk page.
When this work has been completed, you may remove this instance of {{SourceReview}} from the code.


Retrieved from "https://proofwiki.org/w/index.php?title=Non-Zero_Real_Numbers_under_Multiplication_form_Abelian_Group&oldid=523247"