Is $a^2 = b^2$ equivalent to $a=b$?

Question

I need to know if $a^2 = b^2$ is equivalent to $a=b$ so that I can prove that a function is injective. The function is $f: n ∈ Z → n^2 - 1 ∈ Z$ by the way.

Answer 1

No it is not.

If $a^2 = b^2$, we might have

• $a = b$

• $a = -b$

which can be combined under one statement $|a| = |b|$.

Answer 2

it is $a^2-b^2=(a-b)(a+b)$ or from $a^2=b^2$ we get $|a|=|b|$

Answer 3

The function is injective if $f(x)=f(y)$ implies $x=y$. Notice that $f(n)=f(-n)$, but $n\neq -n$ (unless $n=0$), so $f$ is not injective.

Are you sure that you have written the right function or domain? (Perhaps $\mathbb{Z}^+$ instead of $\mathbb{Z}$?)