Kenneth Rosen Edition 7 Exercise 2.1 Question 23 (Page No. 126)

Question

How many elements does each of these sets have where $a$ and $b$ are distinct elements?

1.  $P (\left \{a,b, \left \{a,b \right \} \right \})$
2. $P\left \{ \phi, a, \left \{ a \right \},\left \{ \left \{ a \right \} \right \}\right \}$
3. $P(P(\phi ))$

Answer 1

$1)2^{3}=8$

$2)2^{4}=16$

$3)$Elements in Inner Power Set$=2^{0}=1$

     Again Outer Power Set $2^{1}=2$

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$1)$ In 1 st one elements are $\left \{\phi , \left \{ a \right \},\left \{ b \right \},\left \{ \left \{ a,b \right \} \right \} ,\left \{ a,\left \{ a,b \right \} \right \},\left \{ b,\left \{ a,b \right \} \right \},\left \{ a,b \right \},\left \{ a,b,\left \{ a,b \right \} \right \}\right \}$

$2)$In 2nd set elements are$\left \{ \phi ,\left \{ \phi \right \} ,\left \{ a \right \},\left \{ \left \{ a \right \} \right \},\left \{ \left \{ \left \{ a \right \} \right \} \right \},\left \{ \phi ,a \right \},\left \{ \phi ,\left \{ a \right \} \right \},\left \{ \phi ,\left \{ \left \{ a \right \} \right \} \right \},\left \{ a,\left \{ a \right \} \right \},\left \{ a,\left \{ \left \{ a \right \} \right \} \right \},\left \{ \left \{ a \right \} ,\left \{ \left \{ a \right \} \right \}\right \},\left \{ \phi ,a,\left \{ a \right \} \right \},\left \{ \phi ,a,\left \{ \left \{ a \right \} \right \} \right \},\left \{ a,\left \{ a \right \} ,\left \{ \left \{ a \right \} \right \}\right \},\left \{ \phi ,\left \{ a \right \},\left \{ \left \{ a \right \} \right \} \right \},\left \{ \phi ,a,\left \{ a \right \},\left \{ \left \{ a \right \} \right \} \right \}\right \}$

$3)$ In Power of  Power-Set elements are $\left \{ \phi ,\left \{ \phi \right \} \right \}$