(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Let [tex]n \geq 1[/tex] be a positive integer and let [tex]M_n = \{ 1,...,n \}[/tex] be a set with n elements. Denote by [tex]\mathcal{P} (M_n)[/tex] the set of all subsets of M_{n}. For example [tex]\mathcal{P} (M_2) = \{ \{ \emptyset \}, \{ 1 \}, \{ 2 \}, \{ 1,2 \} \}[/tex].

Show that [tex]C=(\mathcal{P} (M_n) , \cap)[/tex] and [tex]U=(\mathcal{P} (M_n) , \cup)[/tex] each has an identity. Decide whether C and U are groups.

3. The attempt at a solution

For [tex]C=(\mathcal{P} (M_n) , \cap)[/tex], let [tex]a,b,c \in \mathcal{P} (M_n)[/tex]

Associativity: [tex](a \cap b) \cap c = a \cup (b \cap c)[/tex] [tex]\checkmark[/tex]

Identity: is the empty set => [tex]a \cap \emptyset = a[/tex] [tex]\checkmark[/tex]

Inverse: I can't see what's the inverse of this group! for an element a we need an inverse b such that [tex]a \cap b = \emptyset[/tex]. I think this is only true when a & b are completely distinct but I'm not sure...

Similarly [tex]U=(\mathcal{P} (M_n) , \cup)[/tex] satisfies the associativity and I think its identity is also [tex]\emptyset[/tex]. But what is the inverse??

I need help finding the inverses, and please let me know if the rest of my working is correct.

Any help is really appreciated.

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# Homework Help: Group or not?

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