Let w denotes the set of words in the English dictionary. Define the relation R by `R = {(x, y) in W xx W}`, the words x and y have at least one letter in common, then R is

Options

A. Reflexive, not symmetric and transitive
B. Not reflexive, symmetric and transitive
C. Reflexive, symmetric but not transitive
D. Reflexive , symmetric and transitive

Answer

To determine the properties of the relation \( R \) defined on the set of words \( W \) in the English dictionary, we need to analyze whether \( R \) is reflexive, symmetric, and transitive.

### Step 1: Check for Reflexivity
A relation \( R \) is reflexive if every element is related to itself. In this case, for any word \( x \) in \( W \), it has all its letters in common with itself. Therefore, the pair \( (x, x) \) is in \( R \) for every word \( x \).

**Conclusion:** \( R \) is reflexive.

### Step 2: Check for Symmetry
A relation \( R \) is symmetric if whenever \( (x, y) \) is in \( R \), then \( (y, x) \) is also in \( R \). If \( x \) and \( y \) have at least one letter in common, then it follows that \( y \) and \( x \) also have that same letter in common.

**Conclusion:** \( R \) is symmetric.

### 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 \( R \). Here, if \( x \) and \( y \) share at least one letter, and \( y \) and \( z \) share at least one letter, it does not necessarily mean that \( x \) and \( z \) share a letter. For example, consider the words "cat", "bat", and "dog". "cat" and "bat" share the letter 'a', and "bat" and "dog" share no letters. Thus, "cat" and "dog" do not share any letters.

**Conclusion:** \( R \) is not transitive.

### Final Conclusion
The relation \( R \) is reflexive and symmetric but not transitive.

### Summary of Properties
- **Reflexive:** Yes
- **Symmetric:** Yes
- **Transitive:** No