Which of the following statements are true about the sets.

A. 0 EØ

B. Ø E {0}

C. Ø E {Ø}

D. {Ø} ∈ {Ø}

E. {Ø} ⊂ {Ø, {Ø}}

Choose the correct answer from the options given below:

The correct answer is option 4.

Key Points:

A. 0∈∅0∈∅ 0<mo>∈<mi mathvariant="normal">∅</mi></mo> 0∈∅

• Explanation: The empty set (∅∅ ∅ ∅) contains no elements.

• Conclusion: False, because 00 0 0 is not an element of ∅∅ ∅ ∅.

B. ∅∈{0}∅∈{0} ∅<mo>∈<mo stretchy="false">{<mn>0<mo stretchy="false">}</mo></mn></mo></mo> ∅∈{0}

• Explanation: The set {0}{0} {0} has only one element: 00 0 0. The empty set is not listed as an element.

• Conclusion: False, because ∅∅ ∅ ∅ is not in {0}{0} {0}.

C. ∅∈{∅}∅∈{∅} ∅∈{∅}

• Explanation: The set {∅}{∅} {∅} has one element: the empty set itself.

• Conclusion: True, because ∅∅ ∅ ∅ is explicitly an element of {∅}{∅} {∅}.

D. {∅}∈{∅}{∅}∈{∅} {∅}∈{∅}

• Explanation: The set {∅}{∅} {∅} contains only ∅∅ ∅ ∅, not {∅}{∅} {∅} itself.

• Conclusion: False, because {∅}{∅} {∅} is not an element of itself.

E. {∅}⊂{∅,{∅}}{∅}⊂{∅,{∅}} {∅}⊂{∅,{∅}}

• Explanation:

  • {∅,{∅}}{∅,{∅}} {∅,{∅}} has two elements: ∅∅ ∅ ∅ and {∅}{∅} {∅}.

  • The subset {∅}{∅} {∅} is contained within it.

• Conclusion: True, because every element of {∅}{∅} {∅} (which is just ∅∅ ∅ ∅) is also in {∅,{∅}}{∅,{∅}} {∅,{∅}}

Correct Option:

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

Key Concepts:

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

   • Example: ∅∈{∅}∅∈{∅} ∅∈{∅} is true.

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

   • Example: {∅}⊂{∅,{∅}}{∅}⊂{∅,{∅}} {∅}⊂{∅,{∅}} is true.

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

This aligns perfectly with Option 4.