Which of the following cannot have a proper subset ?

Options

A. `{ x : x in Q, 5 lt x lt 7}`
B. `{ x : x in Z, - 4 lt x lt 4}`
C. `{ x : in N, 5 lt x lt 6}`
D. `{ x : x + 1 = 0, x in Z}`

Answer

To solve the question "Which of the following cannot have a proper subset?", we will analyze each option step by step:

### Step 1: Understand Proper Subsets
A proper subset of a set is a subset that contains some but not all elements of the original set. A set that has only one element cannot have a proper subset because the only subset it can have is itself.

### Step 2: Analyze Each Option

**Option A:**
- Set: \( x \) such that \( x \in \mathbb{Q} \) and \( 5 < x < 7 \).
- The numbers between 5 and 7 are rational numbers like 5.1, 5.2, ..., 6.9.
- This set contains infinitely many elements (rational numbers), so it can have proper subsets.

**Option B:**
- Set: \( x \) such that \( x \in \mathbb{Z} \) and \( -4 < x < 4 \).
- The integers in this range are -3, -2, -1, 0, 1, 2, 3.
- This set has several elements, so it can have proper subsets.

**Option C:**
- Set: \( x \) such that \( x \in \mathbb{N} \) and \( 5 < x < 6 \).
- The only natural number that satisfies this condition is none, as natural numbers are whole numbers starting from 1 and do not include decimals.
- Therefore, this set is empty. An empty set has no elements, and thus cannot have a proper subset.

**Option D:**
- Set: \( x \) such that \( x + 1 = 0 \) where \( x \in \mathbb{Z} \).
- The solution to this equation is \( x = -1 \).
- This set contains only one element (-1), which means it cannot have a proper subset.

### Conclusion:
The only option that cannot have a proper subset is **Option C**, as it is an empty set.

### Final Answer:
**Option C** cannot have a proper subset.

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