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The pre-image of a function refers to the set of all input values that are mapped to a given set of output values under a specific function. When you have a function like \( g: \mathbb{N} \times \mathbb{N} \rightarrow \mathbb{N} \) given by \( g(m, n)=2^{m} 3^{n} \), finding the pre-image of a set means identifying all ordered pairs \((m, n)\) such that the function value \( g(m, n) \) is included in the set.

In our exercise, when determining \( g^{-1}(C) \), we're hunting for pairs \( (m, n) \) that satisfy \( g(m, n) \in C = \{1, 4, 6, 9, 12, 16, 18\} \). Here, each ordered pair corresponds to a specific value in \( C \) after applying the function, showcasing their pre-image.
To find the pre-image, consider each element in the set \( C \) and perform trial and error by substituting values of \( m \) and \( n \) into \( g(m, n) \). This allows us to systematically determine the combinations that work.

An inverse function essentially reverses the effect of a function. If you have a function \( f \) that takes \( x \) to \( y \), then the inverse function, denoted as \( f^{-1} \), maps \( y \) back to \( x \). However, not all functions possess an inverse.

In this exercise, the focus is not strictly on one-to-one inverses, but rather on operations that resemble inverse functionality, specifically in terms of pre-images and images like \( g^{-1}(C) \).
Calculating \( g(g^{-1}(C)) \) answers a common inverse-related question. It involves applying the function \( g \) to its pre-image found in the previous step. After calculation, you should retrieve the original set \( C \), demonstrating how a function and its inverse-like operation interact.

Ordered pairs, denoted as \((a, b)\), are foundational in mathematics, especially when dealing with functions. In a Cartesian product, ordered pairs are combinations selected from two sets, where the order matters.

In this example, \( A \times A \) forms ordered pairs such as \((1,2)\), \((2,3)\), and so forth from the set \( A = \{1, 2, 3\} \). Each ordered pair is a potential input for the function \( g(m, n) = 2^m 3^n \).
Ordered pairs are crucial in defining the function \( g \), where the first element contributes to the power of 2, and the second element to the power of 3. Exploring each pair, like in \( g^{-1}(C) \), allows students to appreciate the construction and application of functions across cartesian pairs.

Discrete mathematics is a branch of mathematics dealing with countable, distinct elements. It plays an integral role when examining sets, functions, and ordered pairs.

This exercise involves applying concepts from discrete mathematics, such as Cartesian products, functions, and their inverses, to solve problems like determining \(g(A \times A)\) or finding \(g^{-1}(C)\).

• **Functions**: Used to map elements from one set to another. In this case, mapping ordered pairs to integers.
• **Sets**: We deal with finite sets and their manipulations.
• **Cartesian Products**: Create specific input combinations for functions.

Working with scenarios from discrete mathematics, such as unique mapping and set operations in this problem, builds foundational skills essential for computer science and related fields.