How to verify the axioms for an abelian group?

Question

$$ x \Leftrightarrow y = \begin{cases}1 &: \text{for } (x,y) \in \{(1,1),(0,0)\}\\ 0 &: \text{for } (x,y) \in \{(1,0), (0,1)\}\end{cases} $$

Question: How to verify the axioms for an abelian group?

associative:

$(1,1) \rightarrow (1,1),$ $(0,0) \rightarrow(0,0),$ $(1,0) \rightarrow (0,1),$ $(0,1) \rightarrow (1,0)$

neutral element: $1$

How can I verify if all elements are invertible?

Answer 1

You can find an inverse for each element (which isn't much work in this case). Thus, for every element $x$ , you have to find an element $y$ that acts like an inverse. In quantor notation:

$y = x^{-1} \iff \exists y : x*y = y*x = 1$