Question

If A is null set and B=P(P(P(A))), where P(A) denotes power set of A, Then which of the following is are correct

• A. {} $\subset$ B $\phi \subset$ B
• B. {{}} $\subset$ B {$\phi$} $\subset$ B
• C. {{ }} $\in$ B {$\phi$} $\subset$ B
• D. {} $\in$ B $\phi \in$ B

Answer 1

Answer: (A), (B), (C), (D)

A, B, C, D

Answer 2

To solve this problem, we first need to determine the set B by applying the power set operation iteratively starting from the null set A.

1. Define A: A is a null set, so $A = \phi = \{\}$.

2. Calculate P(A): The power set of A, denoted P(A), is the set of all subsets of A. Since A is the empty set, its only subset is itself (the empty set). $P(A) = \{\phi\}$

3. Calculate P(P(A)): Let $X = P(A) = \{\phi\}$. We need to find the power set of X. The subsets of X are:

   • The empty set: $\phi$
   • The set containing the element $\phi$: $\{\phi\}$ So, $P(P(A)) = \{\phi, \{\phi\}\}$.

4. Calculate P(P(P(A))) (which is B): Let $Y = P(P(A)) = \{\phi, \{\phi\}\}$. We need to find the power set of Y. The subsets of Y are:

   • The empty set: $\phi$
   • The set containing the first element of Y: $\{\phi\}$
   • The set containing the second element of Y: $\{\{\phi\}\}$
   • The set containing both elements of Y: $\{\phi, \{\phi\}\}$ So, $B = P(P(P(A))) = \{\phi, \{\phi\}, \{\{\phi\}\}, \{\phi, \{\phi\}\}\}$.

Now, let's evaluate each given option using the calculated set B. Remember that $\phi$ and {} represent the same null set.

Option A: {} $\subset$ B This statement means $\phi \subset B$. The empty set ($\phi$) is a subset of every set. This is a fundamental property of sets. Therefore, {} $\subset$ B is CORRECT.

Option B: {{}} $\subset$ B This statement means $\{\{\}\} \subset B$. For a set $X$ to be a subset of a set $Y$ ($X \subset Y$), every element of $X$ must be an element of $Y$. Here, $X = \{\{\}\}$. The only element of $X$ is $\{\}$ (which is $\phi$). So, for $\{\{\}\} \subset B$ to be true, we must have {} $\in$ B. From our calculation of B, $B = \{\phi, \{\phi\}, \{\{\phi\}\}, \{\phi, \{\phi\}\}\}$. We can see that $\phi$ (which is {}) is indeed an element of B. Therefore, {{}} $\subset$ B is CORRECT.

Option C: {{ }} $\in$ B This statement means $\{\{\}\} \in B$. For an element $x$ to be an element of a set $Y$ ($x \in Y$), $x$ must be one of the items listed inside the curly braces of $Y$. We need to check if $\{\{\}\}$ (which is $\{\{\phi\}\}$) is an element of B. $B = \{\phi, \{\phi\}, \{\{\phi\}\}, \{\phi, \{\phi\}\}\}$. The third element listed in B is $\{\{\phi\}\}$. Therefore, {{ }} $\in$ B is CORRECT.

Option D: {} $\in$ B This statement means $\phi \in B$. We need to check if $\phi$ is an element of B. $B = \{\phi, \{\phi\}, \{\{\phi\}\}, \{\phi, \{\phi\}\}\}$. The first element listed in B is $\phi$. Therefore, {} $\in$ B is CORRECT.

All the given options are correct based on the definitions of sets, power sets, elements, and subsets. Since the question asks "which of the following is are correct", it implies that multiple options can be correct.

The final answer is $\boxed{A, B, C, D}$

Explanation of the solution:

1. A = $\phi$: The null set.
2. P(A) = $\{\phi\}$: Power set of $\phi$ contains only $\phi$.
3. P(P(A)) = $\{\phi, \{\phi\}\}$: Power set of $\{\phi\}$ contains $\phi$ and $\{\phi\}$.
4. B = P(P(P(A))) = $\{\phi, \{\phi\}, \{\{\phi\}\}, \{\phi, \{\phi\}\}\}$: Power set of $\{\phi, \{\phi\}\}$ contains $\phi$, $\{\phi\}$, $\{\{\phi\}\}$, and $\{\phi, \{\phi\}\}$.
5. Option A ($\phi \subset B$): True, as $\phi$ is a subset of every set.
6. Option B ($\{\{\}\} \subset B$): True, because its only element, $\phi$, is an element of B.
7. Option C ($\{\{\}\} \in B$): True, as $\{\{\phi\}\}$ is explicitly an element in the set B.
8. Option D ($\{\} \in B$): True, as $\phi$ is explicitly an element in the set B.

All options are mathematically correct.