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

[smerhej]

Homework Statement

Establish a \bullet 0 = 0

Homework Equations

A few axioms I thought to be relevant..

The Attempt at a Solution

a = a

a \bullet 0 = 0 \bullet a

(a \bullet 0) - (0 \bullet a) = 0

a(1 \bullet 0) - a(0 \bullet 1) = 0

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

a \bullet 0 = 0

 

[Mark44]

Homework Statement

Establish a \bullet 0 = 0

Homework Equations

A few axioms I thought to be relevant..

The Attempt at a Solution

a = a

a \bullet 0 = 0 \bullet a

(a \bullet 0) - (0 \bullet a) = 0

a(1 \bullet 0) - a(0 \bullet 1) = 0

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

a \bullet 0 = 0

What set does a belong to? What operation does $\bullet$ represent?

What is the actual problem statement?

 

[smerhej]

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.

 

[Mark44]

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.

 

[STEMucator]

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.

 

[Fredrik]

A similar approach is to use that 0=0+0.

a = a

a \bullet 0 = 0 \bullet a

In this attempt, you haven't made it clear how the first equality implies the second.

 

[Dickfore]

\bullet 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:
<br /> x = x + 0, \forall x \in F<br />
Multiply by a, and use the distributive law:
<br /> a \bullet x = a \bullet (x + 0)<br />
<br /> a \bullet x = a \bullet x + a \bullet 0<br />
Use the group properties of addition of the field. What can you say about a \bullet 0 then? Similarly, you can multiply from the left and draw a conclusion for 0 \bullet a.