Let N be the set of nutural numbers. Deseribe the following relation in words giving its domain and the range.
`{(1,1),(16,2),(81,3),(256,4)}`

Answer

To describe the given relation and identify its domain and range, we can follow these steps:

### Step 1: Identify the Relation
The relation provided is:
\[ \{(1,1), (16,2), (81,3), (256,4)\} \]
This means we have pairs of numbers where the first element (x) is related to the second element (y).

### Step 2: Understand the Pattern
Looking at the pairs:
- For \( (1, 1) \): \( 1 = 1^4 \)
- For \( (16, 2) \): \( 16 = 2^4 \)
- For \( (81, 3) \): \( 81 = 3^4 \)
- For \( (256, 4) \): \( 256 = 4^4 \)

From this, we can see that the first element of each pair is the fourth power of the second element.

### Step 3: Describe the Relation in Words
The relation can be described as:
"The first element (x) is the fourth power of the second element (y)."

### Step 4: Identify the Domain
The domain is the set of all first elements (x) in the relation:
\[ \text{Domain} = \{1, 16, 81, 256\} \]

### Step 5: Identify the Range
The range is the set of all second elements (y) in the relation:
\[ \text{Range} = \{1, 2, 3, 4\} \]

### Final Description
Thus, the relation can be summarized as follows:
- **Relation**: The first element (x) is the fourth power of the second element (y).
- **Domain**: \{1, 16, 81, 256\}
- **Range**: \{1, 2, 3, 4\}

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