Proving x=y using field axioms in R | Field Axiom Proof

[Tomp]

Homework Statement

Using only the field axioms, prove that if x,y ε R and x = y then wx = wy.

Homework Equations

http://mathworld.wolfram.com/FieldAxioms.html

The Attempt at a Solution

The solution to this can be solved within 2 lines or so using the field axiom inverses/multiplications ,however, this is one of a couple of assignment questions and it seems a little too easy? Am I thinking about this question correctly?

 

[christoff]

It's a very easy problem - there is no "trick" to it. I just did this quickly to check it; I got it using distributivity and additive inverses.

 

[STEMucator]

Homework Statement

Using only the field axioms, prove that if x,y ε R and x = y then wx = wy.

Homework Equations

http://mathworld.wolfram.com/FieldAxioms.html

The Attempt at a Solution

The solution to this can be solved within 2 lines or so using the field axiom inverses/multiplications ,however, this is one of a couple of assignment questions and it seems a little too easy? Am I thinking about this question correctly?

Notice :

x = y ⇔ x - y = 0

So similarly :

wx = wy
wx - wy = 0
w(x-y) = 0
w0 = 0

So either w=0 or 0=0. I am pretty sure your question should require that w≠0 and then the rest should be obvious.

 

[christoff]

Im pretty sure your question should require that w≠0 and then the rest should be obvious.

Why should you require w≠0? It's still a true statement if you take any w real.

 

[Mark44]

Notice :

x = y ⇔ x - y = 0

So similarly :

wx = wy

NO!
The equation above is what the OP needs to show. You can't start off by assuming what you're trying to prove.

wx - wy = 0
w(x-y) = 0
w0 = 0

So either w=0 or 0=0. I am pretty sure your question should require that w≠0 and then the rest should be obvious.

 

[micromass]

I don't think it can be proven from the field axioms. Rather, you need axioms for the equality operator. This is usually handled in logic courses.

For now, I think it is enough to say that: x and y are the same number, so wx and wy must be the same number as well.

For a more rigorous approach, you need mathematical logic.