**Question:** For any set \( A \), both \(\emptyset\) (the empty set) and \( A \) are in the power set \(\mathcal{P}(A)\).**Options:** - True - False **Explanation:** The statement is asking if both the empty set and the set itself are always included in the power set of \( A \). Remember, the power set \(\mathcal{P}(A)\) is defined as the set of all possible subsets of \( A \). By definition, the empty set \(\emptyset\) and \( A \) itself are always subsets of \( A \), so they are elements of \(\mathcal{P}(A)\).

Answered: For any set A, both Ø and A are in the…

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Advanced Math

For any set A, both Ø and A are in the power set (A).

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Expert Solution

Step 1: Power set

The power set P(A) of a set A is the set of all possible subsets of A, including the empty set (∅) and A. Therefore, by definition, both ∅ and A are in P(A) for any set A.

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