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So, I'm stuck at these questions in set theory.

  1. What is |P({∅,{∅},{∅,{∅}}})|

  2. What is |P({{∅,{∅},{∅,{∅}}}})|

  3. Write every element in P({∅,{∅},{∅,{∅}}}) and in P({{∅,{∅},{∅,{∅}}}}).

I know that the way to find the power set is $ 2^n $ in which n is the number of elements. My guess would therefore be that the the answer to question 1 should be 8 since it contains 3 elements. However, I don't really see any difference between that and the set in question 2. Why are they two different sets?

Regarding question 3, I don't really know where to start there.

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  • $\begingroup$ The first asks about the set which contains the three elements $\varnothing, \{\varnothing\}$ and $\{\varnothing, \{\varnothing\}\}$. The second asks about the set containing the single element $\{\varnothing, \{\varnothing\},\{\varnothing, \{\varnothing\}\}\}$ $\endgroup$ – Arthur Oct 16 '17 at 12:05
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2: Example: Consider the set $\{a, b\}$. This set has two elements: $a$, and $b$. Now, consider the set $\{\{a, b\}\}$. This set has one element, which is "the set containing $a$ and $b$."

3: To write power sets, it may be helpful think of every element of the set as though it were a light switch that you could independently turn on (to indicate that the element should be in the set) or off (to indicate that it's out). The four elements of $P(\{a, b\})$ are $$\{\}, \{a\}, \{b\}, \{a,b\}$$ so if we wanted to write that in set notation, we string the elements together inside brackets: $$P(\{a,b\}) = \{\{\}, \{a\}, \{b\}, \{a,b\}\}.$$ Here, the light switch interpretation of these four elements is: "off and off, on and off, off and on, on and on," respectively. When we consider $P(\{\{a,b\}\})$, we again recall that it is a set with just one element -- so the power set will have just $2^1$ elements, corresponding to whether that one element is in or out. That is, $$P(\{\{a,b\}\}) = \{\{\}, \{\{a,b\}\}\}.$$

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