[Math] How to prove $A=R-\{-1\}$ and $a*b = a+b+ab $ is a binary operation

binary operations

$A=R-\{-1\}$ and $a*b = a+b+ab $

  1. Show that * is a binary operation on A
  2. Show that * is associative
  3. Show that there is an identity element in A for *
  4. Show that every element in A has an inverse with respect to *

I'm new to this lesson. All I know to prove is associativity,

let $a,b,c\in A$

Consider:$$(a*b)*c=(a+b+ab)*(c)$$
$$(a*b)*c=[a+b+ab+c+(a+b+ab)c]$$
$$(a*b)*c=a+b+c+ab+ac+bc+abc$$

Now consider:
$$a*(b*c)=a*(b+c+bc)$$
$$a*(b*c)=[a+(b+c+bc)+a(b+c+bc)]$$
$$a*(b*c)=a+b+c+ab+ab+ac+abc$$
Since we get the same result it is associative.

How to prove part 1? and How do I prove that identity exist? Thanks in advance!

P.S: also I'm curious to know why -1 is omitted in the set?

Best Answer

For the binary operation, you need to prove that $a*b\ne-1$ iff $a$, $b\ne-1$, that is $$a*b+1=a+b+ab+1\ne0.$$

For identity, you want an $e$ with $a*e=e*a=a$. As $*$ is commutative, all one needs is that $a*e=a$, that is $$a+e+ae=a.$$ Can you solve that for $e$ in terms of $a$? And is the result independent of $a$?

Once you have done that, do inverses. You then need to solve $a*b=e$ for $b$ in terms of $a$, that is $$a+b+ab=e$$ where you now know $e$.

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