Exercise · ZW-POWER-SET-008 · ★★☆ Intermediate
Which statement is true for every set \(A\)?
Choose one option.
Show answer
- B. \(\mathcal{P}(A)\) is never empty
Explanation
\(\mathcal{P}(A)\) always contains \(\varnothing\), so it is never empty. Also \(A \subseteq A\) for every \(A\), so \(A\) always belongs to its own power set.
Review the concept: Power set
The set of all subsets of a set \(A\), denoted \(\mathcal{P}(A)\).
\[\mathcal{P}(A) = \{X : X \subseteq A\}\]
More practice problems: power set
- ZW-POWER-SET-001 What is P(A)?
- ZW-POWER-SET-002 How many elements does P(∅) have?
- ZW-POWER-SET-003 How many elements does P(\1,2,3\) have?
- ZW-POWER-SET-004 If X ∈ P(A), what holds?
- ZW-POWER-SET-005 Is ∅ ∈ P(A) for every set A?
- ZW-POWER-SET-006 Let A=\1,2,3,4\. How many elements of P(A) have exactly two elements?
- ZW-POWER-SET-007 What is P(\a\) ∩ \a\?
- ZW-POWER-SET-009 Let A=\a,b\. How many pairs (X,Y) with X,Y ∈ P(A) satisfy X ⊆ Y?