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Chapter 1: Problem 10

If set \(\mathrm{A}\) is empty set then \(\mathrm{n}[\mathrm{P}[\mathrm{P}[\mathrm{P}(\mathrm{A})]]]=\ldots \ldots \ldots \ldots\) (a) 6 (b) 16 (c) 2 (d) 4

Short Answer

Expert verified
The cardinality of the power set of the power set of the power set of an empty set is \[n[P[P[P(\varnothing)]]] = 4\]. So the correct answer is (d) 4.

Step by step solution

01

Understand the Properties of an Empty Set and a Power Set

Firstly, the student should understand that an empty set, also known as a null or void set, is a set that contains no elements. It is unique and is denoted symbolically as {} or \(\varnothing\). Secondly, the power set of any set, denoted as \(P(A)\), is the set of all possible subsets of set \(A\), including both the empty set and \(A\) itself. Thirdly, the cardinality of a set \(n(A)\) is the number of elements in that set.
02

Calculate the Power Set of the Empty Set

Note that the empty set has one subset, which is the empty set itself. Hence, the power set of the empty set can be written as \(P(\varnothing) = \{\varnothing\}\). This means the power set of the empty set is a set that contains one element, which is the empty set.
03

Find the Cardinality of \(P(\varnothing)\)

The cardinality of the power set of the empty set can be calculated as: \(n[P(\varnothing)] = 1\).
04

Calculate the Power Set of \(P(\varnothing)\) and Its Cardinality

Next, the power set of \(P(\varnothing)\) can be calculated. Note that \(P(\varnothing)\) is a set containing one element. It has exactly 2 subsets: the empty set and the set itself, which are \(P[P(\varnothing)] = \{\varnothing, P(\varnothing)\}\). Therefore, the cardinality of this power set is \(n[P[P(\varnothing)]] = 2\).
05

Calculate the Power Set of \(P[P(\varnothing)]\) and Its Cardinality

Finally, the power set of \(P[P(\varnothing)]\) can be calculated. With two elements in \(P[P(\varnothing)]\), it has \(2^2 = 4\) subsets, which are: \(\{}, \{\varnothing\}, \{P(\varnothing)\}, \{ \varnothing, P(\varnothing) \}\). Therefore, the cardinality of the power set of the power set of the power set of an empty set is \(n[P[P[P(\varnothing)]]] = 4\). The correct answer is (d) 4.

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Most popular questions from this chapter

If \(\mathrm{X} \cup\\{3,4\\}=\\{1,2,3,4,5,6\\}\) the which of the following is true (a) Smallest set \(\mathrm{X}=\\{1,2,5,6\\}\) (b) Smallest set \(\mathrm{X}=\\{1,2,3,5,6\\}\) (c) Smallest set \(\mathrm{X}=\\{1,2,3,4\\}\) (d) Greatest set \(\mathrm{X}=\\{1,2,3,4\\}\)

For \(\mathrm{n}, \mathrm{m} \in \mathrm{N} \mathrm{n} / \mathrm{m}\) means that \(\mathrm{n}\) is a factor of \(\mathrm{m}\), the relation/is (a) reflexive and symmetric (b) transitive and symmetric (c) reflexive transitive and symmetric (d) reflexive transitive and not symmetric

If \(\mathrm{A}=\\{1,3,5,7,9,11,13,15,17\\}, \mathrm{B}=\\{2,4, \ldots 18\\}\) and \(\mathrm{N}\) is the universal set then \(\mathrm{A}^{\prime} \cup\left(\mathrm{A} \cup\left(\mathrm{B} \cap \mathrm{B}^{\prime}\right)\right.\) ) is (a) \(\mathrm{A}\) (b) B (c) \(A \cup B\) (d) \(\mathrm{N}\)

Let W denotes the words in the English dictionary. Define the relation \(\mathrm{R}\) by \(\mathrm{R}=\\{(\mathrm{x}, \mathrm{y}) \in \mathrm{W} \times \mathrm{W}\) the ward \(\mathrm{x}\) and \(\mathrm{y}\) have at least one letter in common \(\\}\) Then \(R\) is (a) Not reflexive, symmetric and transitive (b) Reflexive, symmetric and not transitive (c) Reflexive, symmetric and transitive (d) Reflexive, not symmetric and transitive

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