State, whether the following pairs of sets are equivalent or not:
Set of integers and set of natural numbers.

Answer

To determine whether the set of integers and the set of natural numbers are equivalent, we need to follow these steps:

### Step 1: Define the sets
- **Set of Integers (Z)**: This set includes all whole numbers, both positive and negative, as well as zero. It can be represented as:
\[ Z = \{..., -3, -2, -1, 0, 1, 2, 3, ...\} \]

- **Set of Natural Numbers (N)**: This set includes all positive whole numbers starting from 1. It can be represented as:
\[ N = \{1, 2, 3, 4, ...\} \]

### Step 2: Understand equivalent sets
- Two sets are considered **equivalent** if they have the same number of elements, meaning their cardinalities are equal.

### Step 3: Compare the cardinalities
- The set of integers is infinite, as it contains all positive and negative integers, including zero.
- The set of natural numbers is also infinite, but it only includes positive integers starting from 1.

### Step 4: Analyze the nature of infinity
- Although both sets are infinite, they are not equivalent because they do not contain the same elements. The set of integers includes negative numbers and zero, while the set of natural numbers does not.

### Conclusion
Since the set of integers and the set of natural numbers do not have the same elements, they are not equivalent sets.

### Final Answer
The set of integers and the set of natural numbers are **not equivalent**.

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