
The cardinality of the set $P\{P[P(\phi )]\}$ is
A. \[0\]
B. \[1\]
C. \[2\]
D. \[4\]
Answer
163.8k+ views
Hint: To solve this question, we will first calculate the power set of $\phi $ that is $P(\phi )$which is termed as the empty or null set and it contains only one element $\phi $. We will then calculate the power set of $P[P(\phi )]$ and then again calculate the power set of $P[P(\phi )]$ that is $P\{P[P(\phi )]\}$ and determine the number of elements it will contain. We will then count the total number of elements in set $P\{P[P(\phi )]\}$ and determine its cardinality.
Complete step by step solution:We are given a set $P\{P[P(\phi )]\}$ and we have to find its cardinality. We know that cardinality of a power set is the total number of elements which are present in the set. We know that a power set is the set of all its subsets including the void set and is denoted with the symbol $P(S)$ .
We will first find the value of $P(\phi )$.
We know that power set of $\phi $ that is $P(\phi )$contains only one element which is $\phi $. So,
$P(\phi )=\phi $
Let us consider $P(\phi )=\phi $ as $A$ that is $P(\phi )=\phi =A$.
Now power set of $P(\phi )$ that is $P[P(\phi )]$ will be,
$\begin{align}
& P[P(\phi )]=P[A] \\
& =\{\phi ,A\} \\
& =\{\phi ,\{\phi \}\}
\end{align}$
We will now calculate the power set of $P[P(\phi )]$ that is $P\{P[P(\phi )]\}$.
$\begin{align}
& P\{P[P(\phi )]\}=P\{\phi ,\{\phi \}\} \\
& =\phi ,\{\phi \},\{\{\phi \}\},\{\phi ,\{\phi \}\}
\end{align}$
There are four elements in the set of $P\{P[P(\phi )]\}$ so its cardinality will be \[4\].
Option ‘D’ is correct
Note:We can also calculate cardinality with the help of the formula ${{2}^{n}}$ where $n$ is the total number of elements in the set. As $P(\phi )$ is a null set so its cardinality will be ,
$\begin{align}
& P(\phi )={{2}^{n}} \\
& ={{2}^{0}} \\
& =1
\end{align}$
Cardinality of $P[P(\phi )]$ will be,
$\begin{align}
& P[P(\phi )]={{2}^{1}} \\
& =2
\end{align}$
Cardinality of $P\{P[P(\phi )]\}$ will be,
$\begin{align}
& P\{P[P(\phi )]\}={{2}^{2}} \\
& =4
\end{align}$
Complete step by step solution:We are given a set $P\{P[P(\phi )]\}$ and we have to find its cardinality. We know that cardinality of a power set is the total number of elements which are present in the set. We know that a power set is the set of all its subsets including the void set and is denoted with the symbol $P(S)$ .
We will first find the value of $P(\phi )$.
We know that power set of $\phi $ that is $P(\phi )$contains only one element which is $\phi $. So,
$P(\phi )=\phi $
Let us consider $P(\phi )=\phi $ as $A$ that is $P(\phi )=\phi =A$.
Now power set of $P(\phi )$ that is $P[P(\phi )]$ will be,
$\begin{align}
& P[P(\phi )]=P[A] \\
& =\{\phi ,A\} \\
& =\{\phi ,\{\phi \}\}
\end{align}$
We will now calculate the power set of $P[P(\phi )]$ that is $P\{P[P(\phi )]\}$.
$\begin{align}
& P\{P[P(\phi )]\}=P\{\phi ,\{\phi \}\} \\
& =\phi ,\{\phi \},\{\{\phi \}\},\{\phi ,\{\phi \}\}
\end{align}$
There are four elements in the set of $P\{P[P(\phi )]\}$ so its cardinality will be \[4\].
Option ‘D’ is correct
Note:We can also calculate cardinality with the help of the formula ${{2}^{n}}$ where $n$ is the total number of elements in the set. As $P(\phi )$ is a null set so its cardinality will be ,
$\begin{align}
& P(\phi )={{2}^{n}} \\
& ={{2}^{0}} \\
& =1
\end{align}$
Cardinality of $P[P(\phi )]$ will be,
$\begin{align}
& P[P(\phi )]={{2}^{1}} \\
& =2
\end{align}$
Cardinality of $P\{P[P(\phi )]\}$ will be,
$\begin{align}
& P\{P[P(\phi )]\}={{2}^{2}} \\
& =4
\end{align}$
Recently Updated Pages
Geometry of Complex Numbers – Topics, Reception, Audience and Related Readings

JEE Main 2021 July 25 Shift 1 Question Paper with Answer Key

JEE Main 2021 July 22 Shift 2 Question Paper with Answer Key

JEE Main 2025 Session 2: Exam Date, Admit Card, Syllabus, & More

JEE Atomic Structure and Chemical Bonding important Concepts and Tips

JEE Amino Acids and Peptides Important Concepts and Tips for Exam Preparation

Trending doubts
Degree of Dissociation and Its Formula With Solved Example for JEE

Instantaneous Velocity - Formula based Examples for JEE

JEE Main Chemistry Question Paper with Answer Keys and Solutions

JEE Main Reservation Criteria 2025: SC, ST, EWS, and PwD Candidates

JEE Mains 2025 Cut-Off GFIT: Check All Rounds Cutoff Ranks

Lami's Theorem

Other Pages
Total MBBS Seats in India 2025: Government College Seat Matrix

NEET Total Marks 2025: Important Information and Key Updates

Neet Cut Off 2025 for MBBS in Tamilnadu: AIQ & State Quota Analysis

Karnataka NEET Cut off 2025 - Category Wise Cut Off Marks

NEET Marks vs Rank 2024|How to Calculate?

NEET 2025: All Major Changes in Application Process, Pattern and More
