To determine the properties of the relation \( R \) defined on the set \( A \) of human beings in a town, where \( R = \{ (x,y) : x \text{ is younger than } y \} \), we will analyze whether this relation is reflexive, symmetric, or transitive.
### Step 1: Check for Reflexivity
A relation \( R \) is reflexive if for every element \( x \) in the set \( A \), the pair \( (x, x) \) belongs to \( R \). This means that every element must be related to itself.
- For our relation \( R \), we need to check if \( (x, x) \) is in \( R \).
- The statement would be \( x \text{ is younger than } x \), which is false because no one is younger than themselves.
**Conclusion:** The relation \( R \) is **not reflexive**.
**Hint:** To check reflexivity, see if every element relates to itself.
### Step 2: Check for Symmetry
A relation \( R \) is symmetric if whenever \( (x, y) \) is in \( R \), then \( (y, x) \) must also be in \( R \).
- Suppose \( (x, y) \) is in \( R \). This means \( x \text{ is younger than } y \).
- For symmetry, we need to check if \( (y, x) \) is in \( R \), which would mean \( y \text{ is younger than } x \).
- If \( x \) is younger than \( y \), it is not possible for \( y \) to be younger than \( x \) at the same time.
**Conclusion:** The relation \( R \) is **not symmetric**.
**Hint:** To check symmetry, see if the reverse of any related pair is also related.
### Step 3: Check for Transitivity
A relation \( R \) is transitive if whenever \( (x, y) \) is in \( R \) and \( (y, z) \) is in \( R \), then \( (x, z) \) must also be in \( R \).
- Assume \( (x, y) \) is in \( R \) (i.e., \( x \text{ is younger than } y \)) and \( (y, z) \) is in \( R \) (i.e., \( y \text{ is younger than } z \)).
- For transitivity, we need to check if \( (x, z) \) is in \( R \), which means \( x \text{ is younger than } z \).
- If \( x \) is younger than \( y \) and \( y \) is younger than \( z \), then it follows that \( x \) is indeed younger than \( z \).
**Conclusion:** The relation \( R \) is **transitive**.
**Hint:** To check transitivity, see if the relation holds through a chain of related pairs.
### Final Conclusion
Since the relation \( R \) is not reflexive, not symmetric, but is transitive, we conclude that the correct option is:
**Option 4: Neither Reflexive nor Symmetric nor Transitive.**
