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Discrete mathematics forms the backbone of computer science and information theory, where understanding structures such as sequences and functions is essential.

One of the key topics within discrete mathematics is the concept of properly structured or 'well-formed' expressions, which is applicable to numerous fields including linguistics, logic, and computer programming. For instance, when programming, we must ensure our brackets or parentheses, whether it's {} or [], are well-matched and nested correctly to avoid compilation errors.

In the context of our exercise, discrete mathematics helps us understand the structural properties of sequences of parentheses. Through its principles, we can rigorously define what constitutes a 'valid sequence' and apply mathematical reasoning to verify if a given sequence fits the criteria. It's not just about having the same number of opening and closing brackets, but also about the order in which they appear.

Context-free grammar (CFG) is a type of formal grammar that is particularly relevant in the fields of computer science and linguistics for describing the syntax of programming languages and natural languages.

A CFG consists of a set of production rules that describe how to form strings from the language's alphabet that are valid according to the language's syntax. In essence, it provides us with the rules to generate all possible strings - including parentheses sequences - that are considered correct.

In the case of parentheses, a CFG might define a rule such as: A valid sequence can be an empty sequence, or a left parenthesis followed by a valid sequence and then a right parenthesis, or a valid sequence followed by another valid sequence. This recursive definition is perfect for defining nested structures like mathematical expressions or, as in our exercise, sequences of parentheses.

Nested parentheses are a common feature in mathematical expressions, programming, and context-free grammars. They are sets of parentheses which are located inside other parentheses, and they must follow the rules of proper nesting to be considered valid.

An easy way to determine if a series of nested parentheses is valid is by starting from the innermost pair and working outwards, ensuring that each pair is matched. In our textbook example, analyzing '(( )( ))', we see that the inner pairs create standalone units that can be symbolically 'removed' to assess the outer structure.

Correctly nested parentheses will always have a corresponding opening and closing parenthesis, and will not intersect inappropriately. For example, the sequence '(()' is invalid, as the innermost parenthesis doesn't have a corresponding closing partner. In contrast, the sequence in the exercise was indeed valid, as confirmed by our step-by-step analysis.