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The concept of set theory is foundational to modern mathematics. It’s a branch of mathematical logic that deals with sets, which are collections of objects. Think of it as a way of grouping things together based on common characteristics or properties. In the given exercise, we've been provided an equation and asked to find a specific set, denoted as set A.

Understanding how to translate the elements from an equation to form a concrete set is a fundamental skill in set theory. In this case, we identified that set A consists of numbers that make the equation equal to zero, essentially the roots of the equation. Once we find those roots, we can present them as the elements of our set, as seen in the step-by-step solution.

Another important concept is recognizing that sets, by definition, do not contain duplicate elements. This is why even if a set lists an element twice, like in choice (c), it is treated as having only one instance of that element. This concept is crucial when determining if sets are equal or if one is a subset of another.

Equations are at the heart of algebra and are all about finding unknown values that make a statement true. In our example exercise, solving the equation involves finding values of x for which the product of three terms equals zero. This is based on the Zero Product Property, which states if the product of several factors is zero, at least one of the factors must be zero.

Concretely, we have the factors of an equation which are set to zero individually, as seen in the initial steps provided. It’s like unraveling a mystery, where every step is about isolating our 'suspect' variable, x, to find its true value. Once discovered, these values highlight the intersection points with the x-axis on a graph or, in the context of our set A, the elements that belong in our set.

It’s important to confirm each solution back into the original equation to ensure they're valid—this reinforces not only the practice of solving equations but also leads to a better understanding of the equation's behavior and its graphical representation.

Mathematical proofs are rigorously structured arguments that verify the truth of mathematical statements. They are the very backbone of mathematics, ensuring that our conclusions are not just guesses but are based on logical deduction. In the context of the exercise, showing how each option relates to set A constitutes the proof of each answer.

For example, in comparing set A with the other sets like (a) or (d), we’re essentially providing a proof for why they’re not equal to set A or why one is a subset. This process includes an examination of the elements to see if they match or are contained within set A. By doing so, we make logical deductions about their relationships.

Proofs aren't always presented in the form of written 'theorems' and their 'proofs'; they can also be implicit in exercises like these, where each step forms part of the argument leading to the final answer. The clear, logical progression from one step to the next is itself a form of proof.