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Establish a [itex]\bullet[/itex] 0 = 0

  • Thread starter smerhej
  • Start date
  • #1
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Homework Statement



Establish a [itex]\bullet[/itex] 0 = 0

Homework Equations



A few axioms I thought to be relevant..

The Attempt at a Solution



a = a

a [itex]\bullet[/itex] 0 = 0 [itex]\bullet[/itex] a

(a [itex]\bullet[/itex] 0) - (0 [itex]\bullet[/itex] a) = 0

a(1 [itex]\bullet[/itex] 0) - a(0 [itex]\bullet[/itex] 1) = 0

a[(1 [itex]\bullet[/itex] 0) - (0 [itex]\bullet[/itex] 1)] = 0

a [itex]\bullet[/itex] 0 = 0
 

Answers and Replies

  • #2
33,079
4,783

Homework Statement



Establish a [itex]\bullet[/itex] 0 = 0

Homework Equations



A few axioms I thought to be relevant..

The Attempt at a Solution



a = a

a [itex]\bullet[/itex] 0 = 0 [itex]\bullet[/itex] a

(a [itex]\bullet[/itex] 0) - (0 [itex]\bullet[/itex] a) = 0

a(1 [itex]\bullet[/itex] 0) - a(0 [itex]\bullet[/itex] 1) = 0

a[(1 [itex]\bullet[/itex] 0) - (0 [itex]\bullet[/itex] 1)] = 0

a [itex]\bullet[/itex] 0 = 0
What set does a belong to? What operation does ##\bullet ## represent?

What is the actual problem statement?
 
  • #3
20
0
What we know is a is an element of some field F, and the dot is multiplication. What I wrote as the problem is the entire question.
 
Last edited:
  • #4
33,079
4,783
What we know is a is an element of some field F, and the dot is multiplication. What I wrote as the problem is the entire question.
"element of some field F" wasn't in the original post.

Every element in a field has an additive identity, and every nonzero element has a multiplicative identity.
 
  • #5
Zondrina
Homework Helper
2,065
136
So you want to prove :

a0 = 0 in some field F.

So I'm presuming you're allowed to assume the existence of a unique zero element and unique additive inverses.

Then :

a0 = 0 + a0
= (-(a0) + a0) + a0
.....

As much as id like to help you more, the rest should be obvious. Just use the axioms to justify your steps.
 
  • #6
Fredrik
Staff Emeritus
Science Advisor
Gold Member
10,851
406
A similar approach is to use that 0=0+0.

a = a

a [itex]\bullet[/itex] 0 = 0 [itex]\bullet[/itex] a
In this attempt, you haven't made it clear how the first equality implies the second.
 
  • #7
2,967
5
[itex]\bullet[/itex] is a label for the operation of multiplication in a field F. There is an operation of addition in the field, for which the following is true:
[tex]
x = x + 0, \forall x \in F
[/tex]
Multiply by a, and use the distributive law:
[tex]
a \bullet x = a \bullet (x + 0)
[/tex]
[tex]
a \bullet x = a \bullet x + a \bullet 0
[/tex]
Use the group properties of addition of the field. What can you say about [itex]a \bullet 0[/itex] then? Similarly, you can multiply from the left and draw a conclusion for [itex]0 \bullet a[/itex].
 

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