If $A$ is a set with elements $a, \{b\},\{c\}$ then why $\{a,\{b\}\} \nsubseteq P(A)$, where $P(A)$ is the power set of $A$

A textbook on Elementary Set theory shows an example which says that:-

given a set $$A=\{a,\{b\},\{c\}\}$$ find out if the statements are correct - $$\\ a).\ \ \ \{a,\{b\}\}\in P(A) \\ b). \ \ \ \{a,\{b\}\} \subseteq P(A)$$

Definition : Power set of any set $$A$$ is the set of all subsets of $$A$$, including the empty set and $$A$$ itself, i.e. $$\\ P(A) = \{\phi, \{a\},\{\{b\}\}, \{\{c\}\}, {\{a,\{b\}\}}, {\{a,\{c\}}\}, \{\{b\},\{c\}\}, \{a,\{\{b\},\{c\}\}\}\}$$

From the definition it is clear that the option $$a$$ is obviously true but the textbook claims that the second option $$b$$ is incorrect and here I am stuck.

According to the definition $$a, \{b\}\notin P(A)$$, since $$P(A)$$ contains the set that consists of the elements $$a, \{b\}$$ and the set $$\{a, \{b\}\}$$ that contains the elements $$a, \{b\}$$ should be the subset of $$P(A)$$ and that is why the statement $$\{a,\{b\}\}\subseteq P(A)$$, should be true. But somehow it is incorrect.

Would anyone kindly like to mention where am I wrong? Any help is highly appreciated.

• I have found a similar video tutorial youtube.com/watch?v=y8arakl4WNM which also supports the claim of the book. – vbm Aug 3 '19 at 4:08
• What do you mean when you write $a, \{b\}\nsubseteq P(A)$? – Taroccoesbrocco Aug 3 '19 at 4:11
• I mean as elements they are not in $P(A)$ – vbm Aug 3 '19 at 4:14
• So, you should write $a, \{b\} \notin P(A)$, and not: $a, \{b\}\nsubseteq P(A)$ . – Taroccoesbrocco Aug 3 '19 at 4:15

Your textbook is right, $$\{a,\{b\}\} \not\subseteq \mathcal{P}(A)$$. Indeed, $$a$$ is an element of $$\{a,\{b\}\}$$ but it is not an element of $$\mathcal{P}(A)$$ (see the list of the elements of $$\mathcal{P}(A)$$ that you correctly gave), thus it is not true that every element of $$\{a,\{b\}\}$$ is an element of $$\mathcal{P}(A)$$ (in general, given two sets $$B$$ and $$C$$, $$B \subseteq C$$ means that every element of $$B$$ is an element of $$C$$).
So, when you say that "the set $$\{𝑎,\{𝑏\}\}$$ [...] should be the subset of $$\mathcal{𝑃}(𝐴)$$", you are wrong. You can say that $$\{𝑎,\{𝑏\}\}$$ is a subset of $$A$$.