Let w denote the words in the english dictionary. Define the relation R by: R = `{(x,y) in W xx W` | words x and y have at least one letter in common}. Then R is: (1) reflexive, symmetric and not transitive (2) reflexive, symmetric and transitive (3) reflexive, not symmetric and transitive (4) not reflexive, symmetric and transitive

Options

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

Explanation

To determine the properties of the relation \( R \) defined on the words in the English dictionary, we will analyze whether it 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 the dictionary, we need to check if \( (x, x) \) is in \( R \).

- Since any word has all its letters in common with itself, it follows that \( x \) has at least one letter in common with itself. Therefore, \( (x, x) \) is in \( R \) for all \( x \).

**Conclusion**: The relation \( R \) 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 \( (x, y) \) is in \( R \), this means that words \( x \) and \( y \) have at least one letter in common.
- If they share a letter, then the same letter will also be common when we consider the pair \( (y, x) \).

**Conclusion**: The relation \( R \) 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 \( R \).

- Let’s consider \( (x, y) \) and \( (y, z) \) in \( R \). This means \( x \) and \( y \) share at least one letter, and \( y \) and \( z \) share at least one letter.
- However, it is not guaranteed that \( x \) and \( z \) will share any letters.

**Example**:
- Let \( x = \text{"toy"} \), \( y = \text{"boy"} \), and \( z = \text{"shed"} \).
- Here, \( x \) and \( y \) share the letter 'o'.
- \( y \) and \( z \) share the letter 'b'.
- But \( x \) and \( z \) do not share any letters.

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

### Final Conclusion
Based on our analysis:
- \( R \) is reflexive.
- \( R \) is symmetric.
- \( R \) is not transitive.

Thus, the correct answer is option (1): reflexive, symmetric, and not transitive.

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