What is the cardinality of the set $\{1, 1\}$? What is the cardinality of the set $\{1, 1\}$?
My textbook does not mention this.
 A: Remember that the set $A=\left\{1, 2, 3, 4, 9, 13, monkey, house, pig, 2, 6, 1, 4\right\}$ has 3 elements which are named twice, even though they do not have to be.
Think of it like this: Say I have a list of names in a classroom of 5 children. I could write that list as: $N=\left\{Puck, Elise, Mathew, George, Tim\right\}$. It wouldn't matter how often I repeat one of those names, because the same kid would just be named more often. Meaning that the set $N_2=\left\{Puck, Elise, Mathew, George, Tim, Tim, Puck, Puck, Elise\right\} = N$
This can get confusing when you start including sets in your sets. In which case you have to realise that $\left\{10\right\}\neq 10$. 
This should give you plenty of information to help you on your way.
A: Think about it for a minute. Your book does not have to mention this. It's something you can figure out for yourself. 
$\{1,1\}=\{1\}$, because they're both subsets of each other and thus equal and identical. So their cardinalities are equal. 
Alternatively, the function that maps $1$ to $1$ is a bijection of $\{1,1\}$ to $\{1\}$ (check it). Thus they have the same cardinality: $1$. So $\left |\{1,1\} \right |=1$. 
Adam V. Nease
