Discrete Mathematics - Problems with Languages

[prevail]

Let $$\Sigma$$ = {$$\beta$$,x,y,z} where $$\beta$$ denotes a blank, so x$$\beta \neq$$ x, $$\beta \beta \neq \beta$$, and x$$\beta$$y $$\neq$$ xy but x $$\lambda$$y = xy.

Compute each of the following:

1: $$\parallel \lambda \parallel$$
2: $$\parallel \lambda \lambda \parallel$$
3: $$\parallel \beta \parallel$$
4: $$\parallel \beta \beta \parallel$$
5: $$\parallel \beta$$³ $$\parallel$$
6: $$\parallel$$ x $$\beta \beta$$ x $$\parallel$$
7: $$\parallel \beta \lambda \parallel$$
8: $$\parallel \lambda$$ ¹⁰ $$\parallel$$

Uhm.. can someone help me out ? I've tried like 3 days now (without progress). Discrete math sux :P

 

[EnumaElish]

How is || defined?

 

[HallsofIvy]

You said "$\Sigma= {\beta, x, y, z}$ where $\Beta$ is a blank", but what is $\lambda$?