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I am trying to reduce the notation aab*b I believe ab^2 is the answer - can someone confirm I have it correct or give me the correct simplification

Thanks

Jack Klosterman
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    please see https://math.meta.stackexchange.com/questions/5020/mathjax-basic-tutorial-and-quick-reference – Angina Seng Sep 10 '19 at 15:37
  • Nope, $ab^2$ is a short hand for $abb$. If $a,b$ commutes, you can abbreviate $aabb$ as $(ab)^2$. If $a,b$ didn't commute (e.g. they are two generic $n\times n$ matrices), the best you can do is simplify $aabb$ to $a^2b^2$. – achille hui Sep 10 '19 at 15:39
  • "the best you can do..." IMHO $a^2b^2$ is "better" than $(ab)^2$. – fleablood Sep 10 '19 at 15:48
  • Is $$ supposed to be BOOLEAN multiplication. If so then $aa = a$ and $bb = b$ so $aabb = (aa)(bb) =ab$. And it is true that $aabb = ab^2 = a*b$ but that $ab^2$ is not the simplest expression. – fleablood Sep 10 '19 at 23:07

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If we assume (you did not state this) that $*$ is BOOLEAN multiplication then we have $a*a = a$ for any value of $a$ (because $0*0 = 0$ and $1*1 = 1$ and there aren't any other values.

So $a*a*b*b = (a*a)*(b*b) = a*b$ is the simplest expression.

It's true that $a*a*b*b =(a*a)*b^2 = a*b^2$ but as $b^2 = b$ that's not the simplest.

If $*$ is not the boolean multiplication, read on:

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It depends on whether $*$ is commutative and or associative.

$a*a = a^2$ and $b*b=b^2$. Assuming associativity then $a*a*b*b =(a*a)*(b*b) = a^2*b^2$.

Assuming associativity and commutivity then $a*a*b*b = a*b*a*b =(a*b)*(a*b) = (a*b)^2$. But we can not claim this if $*$ is not commutative.

However then notation $a*b^2$ without parenthesis would mean $a*b*b$ and not $a*a *b *b$ so the parenthesis are needed.

(Note: If $*$ is not associative we are pretty much sunk. The expression $a*a*b*b$ could mean $a*(a*(b*b)))$ or $(a*(a*b))*b$ or $(a*a)*(b*b)$ or $(((a*a)*b)*b$ which can all be different things.)

(If $*$ is boolean multiplication then $*$ is both associative and commutative and $a*a*b*b$ can be written as $a^2*b^2=a*b$.)

fleablood
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  • Well, if * is boolean multiplication, then $a^2 = a$. So it can be written as $ab^2$, but I wouldn't just go and state that without justification. – Brian Tung Sep 10 '19 at 22:43
  • Oh.... I didn't consider that. But then I'd claim $a^2 =a$ and $b^2 = b$ so $aab*b=ab$ would be a simpler expression. – fleablood Sep 10 '19 at 23:05
  • Oh indeed! In fact when I saw the title, I assumed it was binary multiplication and that it would therefore simplify to $ab$. – Brian Tung Sep 11 '19 at 01:37
  • Well, I had a brain fart and totally forgot that $a^2 = a$ always.... I was going for the generic associative and commutative multiplication where $a^2 \ne a$ in general. Ooops. – fleablood Sep 11 '19 at 04:51
  • Well, to be careful, we don't know it's binary multiplication... – Brian Tung Sep 11 '19 at 07:21
  • Well, I did chastise the OP for that. – fleablood Sep 11 '19 at 16:10