Let W denote the words in the English dictionary. Define the relation R by `R:{(x,y) in W xx W` such that the words x and y have at least one letter in common) Then R is:

Options

A. reflexive, symmetric and not transitive
B. reflexive, symmetric and transitive
C. reflexive, not symmetric and transitive
D. not reflexive, symmetric and transitive

Answer

To determine the properties of the relation \( R \) defined on the set of words \( W \) in the English dictionary, where \( R = \{(x,y) \in W \times W \mid x \text{ and } y \text{ have at least one letter in common}\} \), we need to check if \( R \) is reflexive, symmetric, and transitive.

### Step 1: Check Reflexivity
A relation \( R \) is reflexive if for every element \( x \in W \), the pair \( (x, x) \) is in \( R \). This means that every word must have at least one letter in common with itself.

- Since any word shares all its letters with itself, \( (x, x) \) is indeed in \( R \) for all \( x \in W \).

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

### Step 2: Check Symmetry
A relation \( R \) is symmetric if whenever \( (x, y) \in R \), then \( (y, x) \) must also be in \( R \). This means if two words share at least one letter, then the relation should hold in both directions.

- If \( (x, y) \in R \), it implies that \( x \) and \( y \) have at least one letter in common. Therefore, \( y \) and \( x \) also share that same letter.

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

### Step 3: Check Transitivity
A relation \( R \) is transitive if whenever \( (x, y) \in R \) and \( (y, z) \in R \), then \( (x, z) \) must also be in \( R \). This means if \( x \) shares a letter with \( y \), and \( y \) shares a letter with \( z \), then \( x \) must share a letter with \( z \).

- Consider the words "cat", "bat", and "dog":
- "cat" and "bat" share the letter 'a' (so \( (cat, bat) \in R \)).
- "bat" and "dog" share no letters (so \( (bat, dog) \notin R \)).
- Therefore, "cat" and "dog" do not share any letters (so \( (cat, dog) \notin R \)).

This example shows that \( R \) is not transitive.

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

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