Simplify the notation a*a*b*b

Question

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

Answer 1

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:

=======

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 $a^2*b^2=a*b$.)