Given a group consisting of the following :
- $\mathbb{G} = (\mathrm{G}, +)$ where :
- $\mathrm{G}$ is the underlying set of the group algebraic structure
- $+$ is an internal binary operation which follows the identities of the group structure
- some identities which axiomatise the group :
- closure : $\forall a,b \in \mathrm{A} \big( (a+b)\in \mathrm{A} \big)$
- associativity : $\forall a,b,c \in \mathrm{A}, \big( (a + b) + c = a + (b + c) \big)$
- identity element : $\forall a \in \mathrm{A}, \exists e \in \mathrm{A} \big( a + e = e + a = a \big)$
- inverse elements : $\forall a \in \mathrm{A}, \exists i \in \mathrm{A} \big( a + i = i + a = e \big)$
My questions are the about the formalization of the results of the proof of uniqueness of identity element vs inverse elements :
- if one makes the proof for identity element, one must prove (usually by contradiction) there is a unique element $e$ in the entire set $\mathrm{A}$ which follows the identification. From the result of this proof, I made this following formal statement : $$\exists! e \in \mathrm{A} \Big( \forall a \in \mathrm{A} \big(a + e = e + a = a\big) \Big) $$ Is this grammatically correct? What are the other ways to formalize this?
- if one makes the proof for inverse elements, one must prove there is a unique element $i$ for each element $a$ which follows the identification such that : $a\in \mathrm{A}, b\in \mathrm{A}| a \neq b, \exists! i_1 \in \mathrm{A}, \exists! i_2 \in \mathrm{A} | \ a + i_1 = i_1 + a = e, b + i_2 = i_2 + b = e | \ i_1 \neq i_2 $, when I tried to generalize this I wrote: $$\forall a \in \mathrm{A} \Big( \exists! i \in \mathrm{A} \big( a + i = i + a = e\big) \Big)$$ Is this grammatically correct? What are the other ways to formalize this?
When I compared $\exists! e \in \mathrm{A} \Big( \forall a \in \mathrm{A} \big(a + e = e + a = a\big) \Big) $ and $\forall a \in \mathrm{A} \Big( \exists! i \in \mathrm{A} \big( a + i = i + a = e\big) \Big)$ I denoted that I interchanged the following predicates : $\exists! x \in \mathrm{E}$ with $\forall y \in \mathrm{E}$ where $x$ can be $i$ or $e$ depending on the identities, hence because I wrote this statement instinctively (I have never taken an academic course on formal logic), I thought that the placement of the "small" predicate inside the "big predicate" change if the "small predicate" is acting on another or is acted on, is this correct? Do you know the word to differentiates the "big predicates" from the "smaller predicates", so predicates which take predicates as there entries?