Prove that the group4( , )Z  is cyclic. Find all its generators

To prove that the group Z+ is cyclic, we need to show that there exists an element in the group that generates all other elements.

Step 1: Definition of the group Z+ The group Z+ is defined as the set of all positive integers, including zero, under the operation of addition.

Step 2: Existence of a generator To prove that Z+ is cyclic, we need to find an element in Z+ that generates all other elements. Let's consider the element 1.

Step 3: Generating all other elements We need to show that by repeatedly adding 1 to itself, we can obtain all other elements in Z+.

Let's take an arbitrary element n in Z+. We can obtain n by adding 1 to itself n times. For example, if n = 5, we can obtain 5 by adding 1 to itself 5 times: 1 + 1 + 1 + 1 + 1 = 5.

Therefore, we have shown that the element 1 generates all other elements in Z+.

Step 4: Conclusion Since we have found an element (1) in Z+ that generates all other elements, we can conclude that Z+ is cyclic.

Step 5: Finding all generators In Z+, any positive integer can serve as a generator. This is because by repeatedly adding the generator to itself, we can obtain all other elements in Z+.

Therefore, all positive integers in Z+ are generators of the group Z+.