Prove that F satisfies all field axioms by method of direct verification

In summary: Thanks!In summary, the homework statement is to prove that a collection of all rational numbers satisfies all of the field axioms.
  • #1
GOsuchessplayer
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0

Homework Statement



Consider the collection F of all real numbers of the form x+y√2, where x and
y are rational numbers. Prove (by direct verification) that F satisfies all the field
axioms (just like R) under the usual addition and multiplication.


Homework Equations



Field axioms: There exist two binary operations, called addition + and
multiplication ∗, such that the following hold:
1) commutativity x + y = y + x, xy = yx
2) associativity x + (y + z) = (x + y) + z, x(yz) = (xy)z
3) distributivity x(y + z) = xy + xz
4) Existence of 0; 1 such that x + 0 = x, 1 · x = x,
5) Existence of negatives: For every x there exists y such that x + y = 0.
6) Existence of reciprocals: For every x ̸= 0 there exists y such that
xy = 1.

The Attempt at a Solution



I just want to make sure I did this right. If you were to prove that a collection F ( a collection of all real #'s of the form x+y[tex]\sqrt{2}[/tex] where x & y are rational numbers ) satisfies all of the field axioms by direct verification, would you just do something like suppose m,n,o belong to F. then m=x1+y1[tex]\sqrt{2}[/tex], etc. and then you just say m+(n+o) = x1+y1[tex]\sqrt{2}[/tex] + ... until you return to m+n+o = (m+n)+o ? And then proceed to do so for all the axioms mentioned above? It's supposed to be really trivial right?
 
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  • #2
You're on the right track. To be completely rigorous, I'd add the steps to verify that the set F is actually closed under addition and multiplication. That is, for any [tex]x,y\in F[/tex]

1. [tex]x+y\in F,[/tex]
2. [tex]xy \in F.[/tex]

Kind of trivial as you say, but proofs are about covering all of the bases.
 
  • #3
fzero said:
You're on the right track. To be completely rigorous, I'd add the steps to verify that the set F is actually closed under addition and multiplication. That is, for any [tex]x,y\in F[/tex]

1. [tex]x+y\in F,[/tex]
2. [tex]xy \in F.[/tex]

Kind of trivial as you say, but proofs are about covering all of the bases.

Just for clarity,

Why is it necessary to claim that the set F is closed under addition and multiplication for any x,y \in F? In other words, what would happen if the set wasn't closed under addition and multiplication, is it possible for the axioms to hold and this to still be true? Or does it mean that it's impossible for the axioms to hold if the set is not closed under addition and multiplication.
 
  • #4
GOsuchessplayer said:
Just for clarity,

Why is it necessary to claim that the set F is closed under addition and multiplication for any x,y \in F? In other words, what would happen if the set wasn't closed under addition and multiplication, is it possible for the axioms to hold and this to still be true? Or does it mean that it's impossible for the axioms to hold if the set is not closed under addition and multiplication.

Closure is usually the first axiom in the definition. I brought it up because it wasn't listed in your problem. If the set wasn't closed under those operations, there'd be no point in verifying any of the other axioms since it couldn't be a field.
 
  • #5
fzero said:
Closure is usually the first axiom in the definition. I brought it up because it wasn't listed in your problem. If the set wasn't closed under those operations, there'd be no point in verifying any of the other axioms since it couldn't be a field.

Alright,

Got it, You've been a big help.
 

FAQ: Prove that F satisfies all field axioms by method of direct verification

1. What is the method of direct verification?

The method of direct verification is a mathematical approach used to prove that a given structure satisfies a set of axioms. It involves systematically verifying each axiom using logical reasoning and mathematical operations.

2. What are field axioms?

Field axioms are a set of mathematical rules that define the properties of a field, which is a mathematical structure that consists of a set of numbers and two binary operations (addition and multiplication).

3. How do you prove that F satisfies all field axioms?

To prove that F satisfies all field axioms, you need to show that every axiom holds true for all elements in the field. This can be done by directly verifying each axiom using the elements and operations defined in the field.

4. Why is it important to prove that F satisfies all field axioms?

Proving that F satisfies all field axioms is important because it ensures that the field follows all the necessary rules and properties to be considered a valid mathematical structure. It also allows us to make accurate and reliable mathematical calculations and proofs using the field.

5. Are there alternative methods to prove that F satisfies all field axioms?

Yes, there are alternative methods such as using the method of indirect verification (also known as proof by contradiction) or using the method of axiomatic systems. These methods may be more efficient or useful in certain situations, but the method of direct verification is the most straightforward and commonly used approach.

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