# How do we prove that cancellation law is hold? [duplicate]

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?

• The first thing that comes to my mind isn‘t adding $(-3)$ on both sides. – Qi Zhu Sep 30 '19 at 3:11
• Are you doubting whether $a=b$ implies $a+c=b+c$? – Angina Seng Sep 30 '19 at 3:14
• 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. – Qi Zhu Sep 30 '19 at 3:16
• @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. – Qi Zhu Sep 30 '19 at 3:18
• – JMoravitz Sep 30 '19 at 3:18

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