If A is {{Φ}, {Φ, {Φ}}}, then the power set of A has how many element? a. 4 b. 2 c. 6 d. 8

Break Down the Problem

1. Identify the set $A$.
2. Determine the elements of $A$.
3. Use the formula for the power set.

Relevant Concepts

• The power set of a set $S$ is the set of all possible subsets of $S$.
• If a set has $n$ elements, the power set has $2^n$ elements.

Analysis and Detail

1. The set $A = \{\varnothing, \{\varnothing\}\}$ has 2 elements:

   • $\varnothing$ (the empty set)
   • $\{\varnothing\}$ (a set containing the empty set)

2. Since $A$ has 2 elements, we apply the formula for the power set: $\text{Number of elements in the power set} = 2^n$ where $n = 2$ (the number of elements in $A$): $2^2 = 4$

Verify and Summarize

• The subsets of $A$ are:
  1. $\varnothing$
  2. $\{\varnothing\}$
  3. $\{\{\varnothing\}\}$
  4. $A$ itself, $\{\varnothing, \{\varnothing\}\}$

Thus, the total number of elements in the power set of $A$ is indeed 4.

Final Answer

The power set of $A$ has 4 elements. The correct choice is a. 4.