Kenneth Rosen Edition 6th Exercise 2.1 Question 21c (Page No. 120)

Question

21 c) How many elements does the given set have where a and b are distinct elements?

 P(P(∅))

Comments

Please ask each question separately- otherwise its difficult to follow..

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https://gateoverflow.in/18518/how-many-subsets-does-p-a-have-if-a-is-null-set?show=18518#q18518

Answer 1

Power Set of a set is a set of all subsets of it, a subset being a set containing elements from the given set.

Now empty set has no element in it- so it has only one subset - which is the empty set itself. So, power set of empty set is

• A set containing empty set - $\{ \emptyset \}$, and its cardinality is 1. 

Now, we apply power set on this set- so we get two subsets - $\{\emptyset\}, \emptyset$. So,
$$P(P(\emptyset)) = \{ \emptyset, \{\emptyset \}\}$$