# Establish a $\bullet$ 0 = 0

## 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

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Mark44
Mentor

## 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?

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:
Mark44
Mentor
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.

Zondrina
Homework Helper
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
Staff Emeritus
Gold Member
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.

$\bullet$ 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:
$$x = x + 0, \forall x \in F$$
Multiply by a, and use the distributive law:
$$a \bullet x = a \bullet (x + 0)$$
$$a \bullet x = a \bullet x + a \bullet 0$$
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 $0 \bullet a$.