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Formal Specification Set Theory

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Hello, I need a help with working with set theory for pre, post conditions and logic conditions for a board game I wrote in Java. The Java part is easy, figuring out to express what I using set theory is difficult.

A brief set theory overview as it relates to documentation Java code would be helpful and specific explanation of these issues -

I.
An example is attached of practice problem. Could you explain the post condition step by step? I get the general idea the expression is showing a loop that ends when their is nothing left in the string then returns number of characters in the string, but I bit confused by the symbols and notations.

II.
Could you explain how I could use they same type of notation in the example above to write a complex logic case such as -

Write a function that searches a [n][n] matrix filled with x's,y's or blank to find if a location in the matrix exists that:

I. there is x or multiple x values with a blank on one side (left/right or top bottom) and a y on the other
II. there is a y or multiple y values with a blank on one side (left/right or top bottom) and a x on the other

Thank you for help, I am sure with a little guidance I can get the hang of this stuff.