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Discrete mathematics is an area of mathematics that is concerned with distinct, separate, and countable objects. Unlike continuous mathematics, which focuses on objects that can vary smoothly, discrete mathematics deals with objects that can only assume distinct, separated values. Topics in discrete mathematics include studying sets, graphs, algorithms, and integers, among others. It's especially used in computer science where digital systems and computing require a discrete setup, with a focus on integers, logic, and networks.

When we consider integers within discrete mathematics, we’re interested in numbers without any fractional or decimal part, which are fundamental to various computational processes and mathematical proofs. This study is pivotal when we look at the problem of proving certain properties of integers, such as whether a given expression will always result in an odd or even integer.

A mathematical proof is a logical argument that establishes the truth of a mathematical statement beyond any doubt. It involves a sequence of logical deductions from known truths or previously established statements, such as axioms, definitions, or theorems. Proofs are the core mechanism through which mathematical truths are verified and certainty is achieved in mathematics.

To prove that an expression concerning integers, like the one in our exercise, is always odd or even, one typically approaches the problem by considering different cases, often separating the analysis for even and odd integers. By demonstrating that the statement holds true for all possible scenarios, we can conclude that the statement is universally true. This is the essence of a mathematical proof.

Integers are either odd or even. An even integer is defined as any integer that can be exactly divided by two, with no remainder, while an odd integer is an integer that cannot be exactly divided by two - it will always have a remainder of one when divided by two. In a more formal algebraic sense, an even integer can be expressed as any integer of the form

Even Integer:

\(2k\) where
\(k\) is any integer. Conversely, an odd integer can be expressed as

Odd Integer:

\(2k + 1\) where
\(k\) is any integer. This representation is crucial when trying to prove statements about integers such as in the provided exercise: showing that for all integers, a certain expression yields an odd result.

Properties of integers form the backbone of many mathematical theories and applications. One of the most basic properties is the dichotomy between odd and even integers, as described above. Another relevant property is the ability to describe the sum, difference, and product of even and odd integers through simple rules:

• The sum or product of two even integers is even.
• The sum or product of two odd integers is odd.
• The sum of an odd and an even integer is odd.
• The product of an odd and an even integer is even.

Using these properties, we can decompose and analyze mathematical expressions to predict their parity (whether they will be odd or even), which is exactly the kind of analysis we apply when we approach a problem like proving that a given integer expression will always yield an odd number, regardless of whether the input integer is odd or even.