https://file.aiquickdraw.com/mathbot/file/1730613785568knsg5qw2.jpg

The problem given in the image defines a set \( G \) as follows:

\[
G = \{ a \in \mathbb{Z} \mid a \text{ is a multiple of 2, or } a \text{ is a multiple of 3} \}.
\]

The question asks: **Is \( G \) a group under the usual addition?** If yes, you are to **prove your answer**.

### Solution Outline

To determine if \( G \) is a group under addition, we must check whether \( G \) satisfies the four group axioms:
1. **Closure**: If \( a, b \in G \), then \( a + b \in G \).
2. **Associativity**: Addition in \( \mathbb{Z} \) is associative, so this property will automatically hold.
3. **Identity**: There exists an element \( 0 \in G \) such that \( a + 0 = a \) for all \( a \in G \).
4. **Inverses**: For every \( a \in G \), there exists an element \( -a \in G \) such that \( a + (-a) = 0 \).

Let me know if you'd like a detailed, step-by-step proof for each property.

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Do you have any questions, or would you like further clarification on any part?

### Further Questions
1. What would change if \( G \) were defined under multiplication instead of addition?
2. Can we find an example of an element in \( G \) and check if it has an inverse in \( G \)?
3. How does closure under addition depend on the multiples of 2 and 3?
4. What would happen if we restricted \( G \) to only multiples of 2?
5. Can you list some examples of elements in \( G \)?

### Tip
When proving closure for subsets defined by conditions like "multiple of 2 or 3," consider testing sums of examples to see if the property holds.

Define the set G = {a ∈ Z | a is a multiple of 2, or a is a multiple of 3}. Is G a group under the usual addition? Prove your answer.

Explore whether the set G = {a ∈ Z | a is a multiple of 2, or a is a multiple of 3} forms a group under the usual addition. This problem examines core concepts in group theory, such as closure, identity, associativity, and inverses, suitable for undergraduate students studying abstract algebra.

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