How to convince a math teacher of this simple and obvious fact?

I have in my presence a mathematics teacher, who asserts that

ab=cd

Implies:

a=c, b=d

She has been shown in multiple ways why this is not true:

12=48

05=0657

For me, these seem like valid (dis)proofs by contradiction, but she isn’t satisfied. She wants a ‘more mathematical’ proof, and I can’t think of any.

I’m worried that if she isn’t convinced, it may be detrimental to some students. Is there another way to systematically demonstrate the untruth of her conjecture?

EDIT: Since the answer which worked was from a comment, but each answer is also very good, I’m upvoting all of them instead of accepting a specific one. Feel free to close this question for being too open if so a moderator desires.

Answer

You can prove that all the numbers are equal 😉

Let’s assume that for all a,b,c,dR, b0, d0 we have

ab=cd implies a=c  and  b=d.

Now take any two numbers, say p and q, and write

pp=qq.

Using claim () we have p=q. For the special case, where one of them equals zero (e.g. q), use 2p2p=p+qp+q.

I hope this helps 😉

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Source : Link , Question Author : user86484 , Answer Author :
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