Generators of the group $(Z,+)$ are :

Question

Accepted Answer

To understand why Option 2 is the correct answer, we need to delve into the concept of generators in the context of the group $(\mathbb{Z}, +)$, where $\mathbb{Z}$ represents the set of all integers and $+$ denotes the operation of addition.

### Explanation for Option 2: $-1$ and $1$ as Generators

In group theory, a generator of a group is an element that can be used to produce every element of the group through the group operation. For the group $(\mathbb{Z}, +)$, this means that a generator is an integer that can be used to express every other integer through addition and subtraction.

- **Using $1$ as a Generator**: Starting from $0$, we can reach any positive integer $n$ by adding $1$ repeatedly. For example, $1 + 1 + 1 = 3$ gives us $3$, and so on. Similarly, we can reach any negative integer by adding the inverse of $1$, which is $-1$. For instance, $0 + (-1) + (-1) = -2$.

- **Using $-1$ as a Generator**: Similarly, starting from $0$, we can reach any negative integer by adding $-1$ repeatedly. For example, $-1 + (-1) + (-1) = -3$ gives us $-3$. To reach positive integers, we add the inverse of $-1$, which is $1$.

Thus, both $1$ and $-1$ can generate all integers through their repeated addition, making them generators of the group $(\mathbb{Z}, +)$.

### Explanation for Why Other Options are Incorrect

- **Option 1: $0$ and $1$**

- **$0$ as a Generator**: The element $0$ cannot be a generator because adding $0$ to itself any number of times will always yield $0$. It cannot produce any other integer, thus failing to generate the entire group $\mathbb{Z}$.

- **$1$ as a Generator**: While $1$ is indeed a generator, the presence of $0$ in this option makes it incorrect because $0$ cannot generate any other integers.

- **Option 3: $0$ and $-1$**

- **$-1$ as a Generator**: While $-1$ is a valid generator, similar to the reasoning above, $0$ cannot generate any integers other than itself. Therefore, this option is incorrect due to the inclusion of $0$.

- **Option 4: Generators do not exist**

- This option is incorrect because, as demonstrated, both $1$ and $-1$ can generate the entire set of integers through addition. Therefore, generators do exist for the group $(\mathbb{Z}, +)$.

In conclusion, Option 2 is the correct answer because both $-1$ and $1$ are capable of generating the entire group of integers through their repeated addition, fulfilling the definition of generators in group theory.