Question:
If $A$ is the void set $\phi$, then $P\left(A\right)$ has just one element $\phi$ i.e. $P[\phi] = \left\{\phi\right\}$. So number of elements of $P\left[P\left(P\left(\phi\right)\right)\right]$ is
If $A$ is the void set $\phi$, then $P\left(A\right)$ has just one element $\phi$ i.e. $P[\phi] = \left\{\phi\right\}$. So number of elements of $P\left[P\left(P\left(\phi\right)\right)\right]$ is
Updated On: Jul 6, 2022
- $4$
- $3$
- $5$
- $2^4$
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The Correct Option is A
Solution and Explanation
We have, $P(\phi) = \left\{\phi\right\}$
$\therefore P\left(P\left(\phi\right)\right) = \left\{\phi, \left\{\phi\right\}\right\}$
$\Rightarrow P\left[P\left(P\left(\phi\right)\right)\right] = \left\{\phi, \left\{\phi\right\}, \left\{\left\{\phi\right\}\right\}, \left\{\phi, \left\{\phi\right\}\right\}\right\}$
Hence, $P\left[P\left(P\left(\phi\right)\right)\right]$ consists of $4$ elements i.e. $n\left[P\left[P\left(P\left(\phi\right)\right)\right]\right] = 4$
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Concepts Used:
Sets
Set is the collection of well defined objects. Sets are represented by capital letters, eg. A={}. Sets are composed of elements which could be numbers, letters, shapes, etc.
Example of set: Set of vowels A={a,e,i,o,u}
Representation of Sets
There are three basic notation or representation of sets are as follows:
Statement Form: The statement representation describes a statement to show what are the elements of a set.
- For example, Set A is the list of the first five odd numbers.
Roster Form: The form in which elements are listed in set. Elements in the set is seperatrd by comma and enclosed within the curly braces.
- For example represent the set of vowels in roster form.
A={a,e,i,o,u}
Set Builder Form:
- The set builder representation has a certain rule or a statement that specifically describes the common feature of all the elements of a set.
- The set builder form uses a vertical bar in its representation, with a text describing the character of the elements of the set.
- For example, A = { k | k is an even number, k ≤ 20}. The statement says, all the elements of set A are even numbers that are less than or equal to 20.
- Sometimes a ":" is used in the place of the "|".