Solution

The correct answer is option 4.

Key Points:

A.  $0\in \mathrm{\varnothing }$

• Explanation: The empty set ( $\mathrm{\varnothing }$ ) contains no elements.

• Conclusion: False, because  $0$  is not an element of  $\mathrm{\varnothing }$ .

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B.  $\mathrm{\varnothing }\in \{0\}$

• Explanation: The set  $\{0\}$  has only one element:  $0$ . The empty set is not listed as an element.

• Conclusion: False, because  $\mathrm{\varnothing }$  is not in  $\{0\}$ .

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C.  $\mathrm{\varnothing }\in \{\mathrm{\varnothing }\}$

• Explanation: The set  $\{\mathrm{\varnothing }\}$  has one element: the empty set itself.

• Conclusion: True, because  $\mathrm{\varnothing }$  is explicitly an element of  $\{\mathrm{\varnothing }\}$ .

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D.  $\{\mathrm{\varnothing }\}\in \{\mathrm{\varnothing }\}$

• Explanation: The set  $\{\mathrm{\varnothing }\}$  contains only  $\mathrm{\varnothing }$ , not  $\{\mathrm{\varnothing }\}$  itself.

• Conclusion: False, because  $\{\mathrm{\varnothing }\}$  is not an element of itself.

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E.  $\{\mathrm{\varnothing }\}\subset \{\mathrm{\varnothing },\{\mathrm{\varnothing }\}\}$

• Explanation:

  • $\{\mathrm{\varnothing },\{\mathrm{\varnothing }\}\}$  has two elements:  $\mathrm{\varnothing }$  and  $\{\mathrm{\varnothing }\}$ .

  • The subset  $\{\mathrm{\varnothing }\}$  is contained within it.

• Conclusion: True, because every element of  $\{\mathrm{\varnothing }\}$  (which is just  $\mathrm{\varnothing }$ ) is also in  $\{\mathrm{\varnothing },\{\mathrm{\varnothing }\}\}$

Correct Option:

Only C and E are true.
Answer: Option 4 (C and E only). 

Key Concepts:

1. $\in$  (Element of): Checks if an item is directly inside a set.

   • Example:  $\mathrm{\varnothing }\in \{\mathrm{\varnothing }\}$  is true.

2. $\subset$  (Subset of): Checks if all elements of one set are in another.

   • Example:  $\{\mathrm{\varnothing }\}\subset \{\mathrm{\varnothing },\{\mathrm{\varnothing }\}\}$  is true.

3. The empty set ( $\mathrm{\varnothing }$ ) is not an element of every set—only when explicitly included (e.g.,  $\{\mathrm{\varnothing }\}$ ).

This aligns perfectly with Option 4.