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Field axiom for proof

  • Thread starter Tomp
  • Start date
  • #1
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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?
 

Answers and Replies

  • #2
123
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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.
 
  • #3
Zondrina
Homework Helper
2,065
136

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. Im pretty sure your question should require that w≠0 and then the rest should be obvious.
 
  • #4
123
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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.
 
  • #5
33,493
5,183
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. Im pretty sure your question should require that w≠0 and then the rest should be obvious.
 
  • #6
22,097
3,280
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.
 

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