Does $ab = b^2 \implies a=b? $

Question

My textbook says that the follow is not true: $ab = b^2 \implies a=b $

However, I cannot find a single case where its not true.

What am I missing? Is it true or false? How do I go about proving these things?

Generally, its not true but in real numbers it is true.True to prove this for matrices — Ashar Tafhim, Sep 02 '16 at 11:13

Siddharth Bhat · Answer 1 · 2016-09-02T11:16:06.857

2

$5 * 0 = 0 * 0$ but $5 \neq 0$.

In general, you need cancellation for that property to hold. Cancellation is a property that allows one to derive the fact that

$$ ab = ac \implies b = c $$

One example is $\mathbb Z - \{0\}$.

edited Sep 02 '16 at 11:16

answered Sep 02 '16 at 11:12

Siddharth Bhat

Cancellation is not equivalent to existence of inverses. – Wojowu Sep 02 '16 at 11:14
Apologies, mixed the two up. $\mathbb Z$ is a perfect example of the phenomenon :) – Siddharth Bhat Sep 02 '16 at 11:15
What about $\mathbb R-{0}$? – Soham Sep 02 '16 at 11:23
Yes, $\mathbb R - {0}$ is an example as well. It is stronger than $\mathbb Z - {0}$ since $\mathbb R$ has a notion of "inverses" that $\mathbb Z$ does not – Siddharth Bhat Sep 02 '16 at 13:20

score 1 · Answer 2 · answered Sep 02 '16 at 11:13

1

i would write $$ab=b^2$$ and this is equivalent to $$0=b(a-b)$$ thus we get $$a=b$$ or $$b=0$$

answered Sep 02 '16 at 11:13

score 0 · Answer 3 · answered Sep 02 '16 at 11:12

0

Try $b=0$. Then $a\in\Bbb{R}\ni\{0\}$. So $a=0$ is not the unique solution and surely, $a$ not necessarily equals to $b$.

answered Sep 02 '16 at 11:12

Mythomorphic

score 0 · Answer 4 · answered Sep 02 '16 at 11:12

0

Consider $a=1$ and $b=0$. We see $ab = (1)(0) = 0 = 0^2 = b^2$ but $a \neq b$.

answered Sep 02 '16 at 11:12

benguin

4 Answers4