Inductive definition of a given language I'm having some difficulties solving a induction task.
Here is the task i'm working on:

Give an inductive definition of the given language below:
$\{a^n,b^n\mid n\in\{0,1,\dots\}\}\setminus\{\Lambda,a,b,aa,bb,aaa,bbb\}$

Now, I'm not sure how to start here, but how can I effectively give an inductive definition of the language above?
 A: $\underline{\phantom{xxxxxx}}$ and $\underline{\phantom{xxxxxx}}$ are in the language. If $X$ is in the language and $\underline{\phantom{xxxxxx}}$ is true, then $\underline{\phantom{xxxxxx}}$ is also in the language.
Edit: A few problems to work up to this:
Find inductive definitions for the following languages:


*

*$\{a^n\mid n\in\{0,1,\dots\}\}$

*$\{a^n\mid n\in\{5,6,\dots\}\}$

*$\{a^n,b^n\mid n\in \{0,1,\dots\}\}$ (Note that this is shorthand for $\{a^n\mid n\in\{0,1,\dots\}\}\cup\{b^n\mid n\in\{0,1,\dots\}\}$.)
A hint on your problem: can you list the six shortest words in the language you're trying to define inductively?
Edit: Answer to warm-up problem 1:
Let $\mathcal L$ be the smallest language $\mathcal L$ such that  $\Lambda\in \mathcal L$, and for each $W\in \mathcal L$,  $Wa\in\mathcal L$.
Note: I don't really know what form you're expected to produce, so here are two.
Full solution 1 (because I am stupid):
Let $\mathcal A$ be the smallest language such that:


*

*$aaaa\in\mathcal A$

*$\forall W\in A: aW\in A$.


Let $\mathcal B$ be the smallest language such that


*

*$bbbb\in \mathcal B$

*$\forall W\in B: bW\in B$.


Then $\mathcal L = A\cup B$.
Full solution 2:
$\mathcal L$ is the smallest language such that:


*

*$aaaa\in\mathcal L$

*$bbbb\in\mathcal L$

*For each $W\in \mathcal L$, if the first symbol in $W$ is $a$, then $aW\in\mathcal L$.

*For each $W\in \mathcal L$, if the first symbol in $W$ is $b$, then $bW\in\mathcal L$.

