Math Problem Statement

What is the value of P(P(∅)), where P is the power set and ∅ is the empty set?

Solution

The expression provided is P(P())P(P(\emptyset)), where PP denotes the power set and \emptyset represents the empty set.

Let's solve it step by step:

  1. Find P()P(\emptyset):

    • The power set P(S)P(S) of a set SS is the set of all subsets of SS.
    • The empty set, \emptyset, has only one subset: itself. Thus, the power set of the empty set is P()={}P(\emptyset) = \{ \emptyset \}.
  2. Find P(P())P(P(\emptyset)):

    • Now we need to find the power set of {}\{ \emptyset \}.
    • The set {}\{ \emptyset \} has two subsets: the empty set \emptyset and the set itself {}\{ \emptyset \}.
    • Therefore, P({})={,{}}P(\{ \emptyset \}) = \{ \emptyset, \{ \emptyset \} \}.

Hence, the final result is: P(P())={,{}}.P(P(\emptyset)) = \{ \emptyset, \{ \emptyset \} \}.

Would you like further details or have any questions?

Here are 5 related questions to explore:

  1. What is the power set of a set with three elements?
  2. How many elements does the power set of a set with nn elements contain?
  3. What is the relationship between a set and its power set in terms of cardinality?
  4. How does the concept of power sets apply to infinite sets?
  5. Can the power set of a set contain elements other than sets?

Tip: The power set of a set with nn elements always contains 2n2^n elements.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Set Theory
Power Sets

Formulas

The power set of a set S, P(S), is the set of all subsets of S.

Theorems

Power set theorem: The power set of a set with n elements contains 2^n elements.

Suitable Grade Level

Grades 9-12