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\betay \neq xy but x \lambday = 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 \beta³ \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?