- A relation is a set of ordered pairs \((x, y)\).
- The domain is the set of all unique first coordinates (\(x\)-values) from these pairs.
- Extracting these from \(\{ (-4, 4), (8, -1), (9, -9), (1, 2), (1, 8) \}\) gives \(\{ -4, 8, 9, 1, 1 \}\).
- Removing duplicates and ordering yields the final domain: \(\boxed{\{-4, 1, 8, 9\}}\).
Explanation
- Analysis of the Relation and Domain Definition
A relation is a set of ordered pairs. The domain of a relation is defined as the set of all the first coordinates (often called the x-coordinates) of the ordered pairs in the given set. Given the relation \(S = \{(-4, 4), (8, -1), (9, -9), (1, 2), (1, 8)\}\), our goal is to identify all unique first elements from these pairs and list them as the domain.
- Extracting First Coordinates
Looking at the ordered pairs in the set \(S\):
- The pair \((-4, 4)\) has a first coordinate of \(-4\).
- The pair \((8, -1)\) has a first coordinate of \(8\).
- The pair \((9, -9)\) has a first coordinate of \(9\).
- The pair \((1, 2)\) has a first coordinate of \(1\).
- The pair \((1, 8)\) has a first coordinate of \(1\).
The set of first coordinates is \(\{-4, 8, 9, 1, 1\}\).
- Determining the Domain and Finalizing the Answer
To find the domain, we must list only the unique values from the set of first coordinates \(\{-4, 8, 9, 1, 1\}\). Removing the duplicate \(1\), we are left with \(\{-4, 8, 9, 1\}\). It is standard convention to present the elements of a set in ascending order: \(\{-4, 1, 8, 9\}\). Comparing this result with the provided options, we see that it matches the second option.
Examples
In real-world data analysis, identifying the domain of a relation is similar to listing all unique users who accessed a specific server from a log file. If you have a list of entries like (user1, pageA), (user2, pageB), (user1, pageC), the set of unique users is {user1, user2}. This helps in understanding the scope of distinct entities involved in an interaction, just as finding the domain helps us understand the distinct input values for a function or relation.