- #1

mathmari

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MHB

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I want to check the following sets with the corresponding relations if they satisfy the axioms of groups.

- $M=\mathbb{R}\cup \{\infty\}$ with the relation $\min:M\times M\rightarrow M$. It holds that $\min (a, \infty)=\min (\infty, a)=a$ for all $a\in M$.
- $M=n\mathbb{Z}=\{n\cdot a\mid a\in \mathbb{Z}\}$ for a $n\in \mathbb{N}$ with the usual multiplication of integers as the relation.
- $M=\{w,f\}$ with the relation $A\star B:=(A\implies B)$.
- $M=\{g\in G\mid \forall h \in G: g\star h=h\star g\}$ for a group $(G, \star )$ with the relation of the group $(G, \star)$ as the relation.

- Associativity : For all $a, b, c \in G$ it holds that $a \star (b \star c) = (a \star b) \star c$.
- There is an element $e \in G$ with the property $e \star a = a$ for all $a \in G$. ($e$ is the identity of $G$).
- For each element $a \in G$ there is $a' \in G$, such that $a'\star a = e$. ($a'$ is the inverse to $a$).

- The associativity is satisfied, isn't it? Do we jusify that as follows?

$\min (a, \min (b, c)) = \text{ minum of } \{a, b, c\} = \min(\min (a,b), c)$

The identity is $e=\infty$.

In this case there is no inverse. - The associativity is satisfied, as the multiplication of integers is associative.

There is no identity and there is not an inverse for each element (for example there is no inverse for the element $n$). - Could you give me a hint for this set?

- Since we have the relation of the group $(G, \star)$, all the three axioms are satisfied.