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.

A milk bottle has a lower specification of 4.0 liters and a standard deviation of 0.02. The data is normally distributed. If 2.5% of the data is to be below the lower specification, where should the center be located?

Undergraduate senior level Real Analysis.
Please show me formal math proofs.
Show that Q(set of rational numbers) ~ N(set of natural numbers).
(Suggestion from my professor: prove by letting set Qn < Q and a/b (a and b are some integers, but b is not zero) and Q = union of Qn (Qn = Q1, Q2, Q3, ....) .)

Formal Math Proofs
Prove that each of the following sets is countable:
a) The set of all numbers with two distinct decimal expansions (like 0.500... and 0.4999...);
b) The set of all rational points in the plane (i.e., points with rational coordinates);
c) The set of all rational intervals (i.e., intervals with ratio

Bridge to Abstract Mathematics
SetTheory: Set Operations
Let A, B, C and D be sets. Prove that....
a) A B iff A \ B = Ø.
b) If A B U C and A ∩ B = Ø, then A C.
c) C A ∩ B iff C A and C B.
d) If A B, then A \ C B \ C.
e) (A \ B) \ C = (A \ C) \ (B \ C).
f) If A C and B C, then A U B C.
g) (A

Undergraduate senior level Real Analysis.
Please show me formal math proofs.
Prove that the set of all real functions...defined on a set M is of power greater than the power of M...(see attached)

Implement the following specification for an integer function in the client program that returns the number of items in a queue. The queue is unchanged.
int GetLength(QueType queue)
Function: Determines the number of items in the queue.
Precondition: queue has been initialized.
Postconditions: queue is unchanged.
Functio

Given U = {15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25}, A = {16, 18, 20, 22}, and B = {17, 19, 20, 23, 24}.
Find A′ B′. Do not skip to the answer. Show your work, finding A', then B' then find their union.