Exercise · ZW-POWER-SET-009 · ★★★ Advanced
Let \(A=\{a,b\}\). How many pairs \((X,Y)\) with \(X,Y \in \mathcal{P}(A)\) satisfy \(X \subseteq Y\)?
Choose one option.
Show answer
- A. 9
Explanation
Each element of \(A\) can be in neither set, only in \(Y\), or in both, because \(X \subseteq Y\) rules out being only in \(X\). That gives 3 options per element, so \(3^2=9\).
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-008 Which statement is true for every set A?