Identify whether the given pair consists of equal or equivalent but not equal sets or none:
J= ` { x, x in N , x ne x } and K= {x: x in N , 6 lt x lt 7 }.`

Answer

To determine whether the sets \( J \) and \( K \) are equal, equivalent but not equal, or neither, we will analyze each set step by step.

### Step 1: Analyze Set \( J \)

Set \( J \) is defined as:
\[ J = \{ x \,|\, x \in \mathbb{N}, x \neq x \} \]

**Explanation:**
- The condition \( x \neq x \) is never true for any natural number \( x \).
- For any natural number (1, 2, 3, ...), it is always equal to itself.

**Conclusion:**
- Therefore, set \( J \) contains no elements.
- We can conclude that:
\[ J = \emptyset \] (the empty set)

### Step 2: Analyze Set \( K \)

Set \( K \) is defined as:
\[ K = \{ x \,|\, x \in \mathbb{N}, 6 < x < 7 \} \]

**Explanation:**
- The condition \( 6 < x < 7 \) means we are looking for natural numbers that lie strictly between 6 and 7.
- The only natural numbers are whole numbers (1, 2, 3, ...), and there are no whole numbers between 6 and 7.

**Conclusion:**
- Therefore, set \( K \) also contains no elements.
- We can conclude that:
\[ K = \emptyset \] (the empty set)

### Step 3: Compare Sets \( J \) and \( K \)

Now we have:
- \( J = \emptyset \)
- \( K = \emptyset \)

**Conclusion:**
- Since both sets are empty, they contain the same elements (which is none).
- Therefore, we can say that:
\[ J = K \]

### Final Answer

The sets \( J \) and \( K \) are **equal sets** because they both contain the same elements (none).

---