State for each of the following relations whether it is function or not. (Write Yes or No)
`{(x,y):x in A,yinB,y" is the area of a square of side x"}` where A is the set of measurements of length.

Answer

To determine if the given relation is a function, we need to analyze the relationship defined by the set:

\[ R = \{(x, y) : x \in A, y \in B, y \text{ is the area of a square of side } x\} \]

where \( A \) is the set of measurements of length.

### Step 1: Define the Relation
The relation states that for each \( x \) in set \( A \), there is a corresponding \( y \) in set \( B \) such that \( y \) is the area of a square with side length \( x \). The area \( y \) can be mathematically expressed as:

\[ y = x^2 \]

### Step 2: Identify the Sets
Let’s clarify what sets \( A \) and \( B \) represent.

- Set \( A \) (measurements of length) can include any positive real numbers representing lengths (e.g., \( A = \{x_1, x_2, x_3, \ldots\} \)).
- Set \( B \) (areas) will consist of the corresponding areas calculated from the lengths in set \( A \) (e.g., \( B = \{x_1^2, x_2^2, x_3^2, \ldots\} \)).

### Step 3: Check the Function Criteria
For a relation to be a function, each input (or value from set \( A \)) must correspond to exactly one output (or value in set \( B \)).

- For any given \( x \) in \( A \), the area \( y \) is calculated as \( y = x^2 \).
- This means that for every specific value of \( x \), there is only one corresponding value of \( y \).

### Step 4: Conclusion
Since each \( x \) in set \( A \) maps to exactly one \( y \) in set \( B \), we conclude that the relation is indeed a function.

**Final Answer: Yes, it is a function.**

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