Cancellation Law, If $ax=ay$ then $x=y$

Question

It's about cancellation law.

if $ax=ay$ then $x=y$, provided that $a \neq 0$

It says that this only true if it applied in whole integers not just positive integers, I'm not really sure, if I put some numbers it turns that $x \neq y$ after all.

Thanks in advance

It's weird question. The cancellation law stated above is true for real and even complex numbers. If you think it fails sometimes, have you tried to find a counterexample to disprove it? — Kaster, Mar 01 '13 at 12:51
Really sorry, I'm not reading the text carefully, the statement above is correct, my mistake. — Adam, Mar 01 '13 at 12:58

score 0 · Accepted Answer · answered Mar 01 '13 at 13:05

0

$ax=ay$ so $a(x-y)=0$, so because $a\neq 0$ we get $x=y$.

This is true whenever $a\cdot b = 0 \Rightarrow a = 0 $ or $b = 0$, so it works for positive integers as well as for negative integers.

answered Mar 01 '13 at 13:05

Stefan

3,451

Thanks, I can understand more from you answer. – Adam Mar 01 '13 at 13:07
A good observation is that $a.b=0 \implies a=0$ or $b=0$ is actually equivalent to $ax=ay \implies x=y$ if $a\neq 0$ – Aloizio Macedo Mar 01 '13 at 13:58

Cancellation Law, If $ax=ay$ then $x=y$

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