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Functions form a foundational concept in mathematics, especially in discrete mathematics. A function is a rule that assigns each element from one set, called the domain, to exactly one element in another set, known as the codomain. This process of mapping is fundamental as it creates relationships between different sets of numbers or objects.

• The domain is the set of "input" or argument values for which the function is defined. In this exercise, the domain, set \( S \), is \( \{1, 2, 3, 4, 5\} \).
• The codomain is the set of "output" values potentially produced by the function. Here, the codomain is the set of integers \( Z \).
• A key requirement is that for every element in the domain, there should be a corresponding and unique element in the codomain based on the function's rule.

In practical terms, think of a function as a "machine" where you input numbers and the machine provides an output according to certain rules. Each process maintains the principle of mapping precisely, ensuring there's no ambiguity in the function's outcome.

Now that we understand what a function is, let's talk about one-to-one functions, sometimes called injective functions. The concept of one-to-one is all about uniqueness in mappings.
For a function to be one-to-one, each element of the domain should map to a unique element in the codomain. Simply put:

• No two different elements in the domain share the same output value in the codomain.
• If \(f(x_1) = f(x_2)\), then it must be that \(x_1 = x_2\). This ensures that there's a distinct pairing between each element of the domain and its mapped element in the codomain.

In our exercise, after listing and examining the pairs, we find that both \( (2, 5) \) and \( (5, 5) \) map to the same integer 5 in the codomain. This overlap means the function \( f \) is not one-to-one, as it fails the test of uniqueness. Knowing whether a function is one-to-one can be crucial for understanding the behavior of mathematical models and systems.

Set theory is a branch of mathematical logic that studies sets, which are collections of objects. It's a fundamental language in mathematics, allowing us to clearly define, organize, and manipulate different collections of items.

• Sets can be finite, like our set \( S = \{1, 2, 3, 4, 5\} \), or infinite, such as \( Z \), the set of all integers.
• A set is often represented by listing its elements inside curly brackets, as shown above.
• The Cartesian product \( S \times Z \) defines all possible ordered pairs where the first element is from \( S \) and the second is from \( Z \).

In the context of this exercise, expressing a function as a subset of \( S \times Z \) means identifying the specific pairs that the function's rule creates. Considering and understanding these pairs is essential not just for knowing the function’s behavior, but also for validating properties like being one-to-one.
Set theory provides a robust framework for discussing such mappings, making it indispensable in discrete mathematics and various other fields.