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Suppose i have

$$2x+8=3$$

I believe the first think that comes out from your mind is adding $(-3)$ on both sides.

Suppose we don't know about cancellation law. How do you prove that when we adding the same number on both side won't change the equality?

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    $\begingroup$ The first thing that comes to my mind isn‘t adding $(-3)$ on both sides. $\endgroup$ – Qi Zhu Sep 30 '19 at 3:11
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    $\begingroup$ Are you doubting whether $a=b$ implies $a+c=b+c$? $\endgroup$ – Angina Seng Sep 30 '19 at 3:14
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    $\begingroup$ Yes, I can see, it‘s a good question. I believe this depends on the definition of the equal sign - which is probably rigorously done in ZFC with the axiom of extensionality. Hopefully, an expert in set theory/logic will be able to help, though. $\endgroup$ – Qi Zhu Sep 30 '19 at 3:16
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    $\begingroup$ @LordSharktheUnknown I believe the question should rather more be about the foundations of the equal sign, so ultimately the definition/axioms of $=$ which imo is a good “doubt”/curiosity. $\endgroup$ – Qi Zhu Sep 30 '19 at 3:18
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    $\begingroup$ Also related: Is there a low that you can add or multiply to both sides of an equation? $\endgroup$ – JMoravitz Sep 30 '19 at 3:18
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Binary operations are well-defined functions.

Note that $$x=a \iff x+b=a+b$$ and if $a\ne 0$, the $$ax=b \iff x=b/a$$

The reason is that addition, subtraction, multiplication and division by a non-zero element are well-defined functions.

For example if I have $x=a$, since adding $b$ is a function, we get $x+b=a+b$ that is if inputs are the same then outputs are the same.

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