help with set example

Question

A given set: k= {L⊆ {0,1}* s.t. for all w∈L |w|≤ 3}

What is the plot of this set? Is this correct? {{0,1,00,01,10,11,000,001,010,011,100,101,110,111}}

Answer 1

Try to analyse the conditions for $k$. It contains all languages $L$ (over the alphabet $\{0,1\}$), which only contain words of length less than or equal to $3$.

For example $L_{empty}=\{\}$ is an element of $k$. Also $L_1=\{0,1\}$ is an element of $k$. Don't forget that languages can have an empty word, usually denoted $\varepsilon$, so $L_{\leq 2}=\{\varepsilon, 0,1,00,01,10,11\}$ is another example of an element in $k$.

Do you see the pattern now? Can you write down the entire set $k$ (yes it is a set containing other sets)?

If you are familiar with power sets, there is another nice way to write $k$. Think of what the "largest" element in $k$ is and what all the other elements have in common...