# 412.x4 Determine which of the folllowing operations are associative

• MHB
• karush
In summary, the term "associative" refers to operations where the grouping of numbers does not affect the result. To determine if an operation is associative, you can perform it in two different ways and see if the result is the same. Examples of associative operations include addition, multiplication, and exponentiation. An operation cannot be both associative and non-associative, and it is important to determine if an operation is associative because it allows for simplification of mathematical expressions and can be useful in complex equations and mental calculations.
karush
Gold Member
MHB
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

Last edited:

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 $ab \equiv 1 \pmod 6$.

## 1. What does the term "associative" mean in relation to operations?

The term "associative" refers to the property of an operation where the grouping of numbers does not affect the result. In other words, when performing an associative operation on a set of numbers, the order in which the numbers are grouped does not matter.

## 2. How can I determine if an operation is associative?

To determine if an operation is associative, you can perform the operation on a set of numbers in two different ways: first, grouping the numbers in one order, and then grouping them in a different order. If the result is the same, then the operation is associative.

## 3. What are some examples of associative operations?

Some common examples of associative operations include addition, multiplication, and exponentiation. For example, (2 + 3) + 4 = 2 + (3 + 4), (2 * 3) * 4 = 2 * (3 * 4), and (2^3)^4 = 2^(3^4).

## 4. Can an operation be both associative and non-associative?

No, an operation cannot be both associative and non-associative. An operation is either associative or non-associative, depending on whether the grouping of numbers affects the result or not.

## 5. Why is it important to determine if an operation is associative?

Determining if an operation is associative is important because it helps us to simplify mathematical expressions by changing the grouping of numbers without changing the result. This can be especially useful when working with complex equations or performing mental calculations.

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