Question

Let Z={a,c,{a,b}}. What is |Z|?

Answer 1

$Z=\{a,c,\{a,b\}\}$

$\boxed{|Z|=3}$ (treat $\{a,b\}$ as one element of $Z$)

The power set of $Z$ is

$\boxed{2^Z=\bigg\{\{\},\{a\},\{c\},\big\{\{a,b\}\big\},\{a,c\},\big\{a,\{a,b\}\big\},\big\{c,\{a,b\}\big\},\big\{a,c,\{a,b\}\big\}\bigg\}}$

1. $\{a,c\}\subseteq Z$ is true because both $a\in Z$ and $c\in Z$.

2. $a\in Z$ is true.

3. $\{c\}\subseteq Z$ is true (same reason as part 1).

4. $\{c\}\in Z$ is false because $Z$ does not contain the set $\{c\}$, rather just the element $c$ itself.

5. $b\in Z$ is false because the element $b$ on its own simply is not in $Z$. That $b\in\{a,b\}$ does not mean $b\in Z$, but that $b$ belongs to a subset of $Z$.

6. $\{a,b\}\in Z$ is true.