[tex]Z=\{a,c,\{a,b\}\}[/tex]
[tex]\boxed{|Z|=3}[/tex] (treat [tex]\{a,b\}[/tex] as one element of [tex]Z[/tex])
The power set of [tex]Z[/tex] is
[tex]\boxed{2^Z=\bigg\{\{\},\{a\},\{c\},\big\{\{a,b\}\big\},\{a,c\},\big\{a,\{a,b\}\big\},\big\{c,\{a,b\}\big\},\big\{a,c,\{a,b\}\big\}\bigg\}}[/tex]
1. [tex]\{a,c\}\subseteq Z[/tex] is true because both [tex]a\in Z[/tex] and [tex]c\in Z[/tex].
2. [tex]a\in Z[/tex] is true.
3. [tex]\{c\}\subseteq Z[/tex] is true (same reason as part 1).
4. [tex]\{c\}\in Z[/tex] is false because [tex]Z[/tex] does not contain the set [tex]\{c\}[/tex], rather just the element [tex]c[/tex] itself.
5. [tex]b\in Z[/tex] is false because the element [tex]b[/tex] on its own simply is not in [tex]Z[/tex]. That [tex]b\in\{a,b\}[/tex] does not mean [tex]b\in Z[/tex], but that [tex]b[/tex] belongs to a subset of [tex]Z[/tex].
6. [tex]\{a,b\}\in Z[/tex] is true.
