412.x4 Determine which of the folllowing operations are associative

[karush]

Determine which of the folllowing operations are Associative, Identity, or Inverse
$\textit{Additive on $\mathbb{Z}$}$
$\textit{Subtraction on $\mathbb{N}$}$
$\textit{Division on $\mathbb{R}$}$
$\textit{Division on $\mathbb{Z}/\{0\}$}$
$\textit{Composition on $D_4$}$
$\textit{Composition on the set of rotations done on $D_4$}$
$\textit{Multiplication mod 6 on $\mathbb{Z}_6$}$

I know this
Associativity: $a(bc)=(ab)c$

Identity: $\exists e \in G$ such that $ ae=ea=a, \forall a \in G$

Inverse: $\forall a \in G \ni ab =ba = c$

we are going thru this next but trying to some understanding on here first
much thanks ahead

 

[jvicens]

After analyzing each operation, here are the results:

1. Additive on $\mathbb{Z}$:
- Associative: Yes
- Identity: 0
- Inverse: For any integer $a$, the additive inverse is $-a$.

2. Subtraction on $\mathbb{N}$:
- Associative: No
- Identity: N/A
- Inverse: N/A (not all natural numbers have an inverse under subtraction)

3. Division on $\mathbb{R}$:
- Associative: No
- Identity: 1
- Inverse: For any real number $a$, the multiplicative inverse is $\frac{1}{a}$.

4. Division on $\mathbb{Z}/\{0\}$:
- Associative: No
- Identity: N/A (not all elements have an identity under division)
- Inverse: For any non-zero integer $a$, the multiplicative inverse is $\frac{1}{a}$.

5. Composition on $D_4$:
- Associative: Yes
- Identity: The identity element in $D_4$ is the identity rotation, which is the rotation that does not change any element in the group.
- Inverse: For any rotation $r$ in $D_4$, the inverse is the rotation that "undoes" $r$.

6. Composition on the set of rotations done on $D_4$:
- Associative: Yes
- Identity: The identity element in this set is the identity rotation, which is the rotation that does not change any element in the group.
- Inverse: For any rotation $r$ in this set, the inverse is the rotation that "undoes" $r$.

7. Multiplication mod 6 on $\mathbb{Z}_6$:
- Associative: Yes
- Identity: 1
- Inverse: For any integer $a$ that is relatively prime to 6, the multiplicative inverse is the integer $b$ such that $ab \equiv 1 \pmod 6$.