Question

Prove the following statements. These exercises are cumulative, covering all techniques addressed in Chapters \(4-7\). Suppose \(a, b \in \mathbb{Z} .\) Prove that \((a-3) b^{2}\) is even if and only if \(a\) is odd or \(b\) is even.

Short answer

The statement is proven. If \(a\) is odd or \(b\) is even, then \((a-3)b^2\) is even. Conversely, if \((a-3)b^2\) is even, then \(b\) must be even.

Step-by-step solution

Part 1: Proving the forward implication

Suppose that \(a\) is odd or \(b\) is even. If \(a\) is odd, \(a-3\) is even because the difference of two odd numbers is even. And obviously if \(b\) is even then \(b^2\) is also even. Since the product of two even numbers is even, then \((a-3)b^2\) is even in either case.

Part 2: Proving the reverse implication

Now suppose \((a-3) b^2\) is even. If \(b\) is odd, then \(b^2\) is odd, and an even number \((a-3) b^2\) can't be the product of two odd numbers. So \(b\) must be even. But if \(b\) is even and \((a-3) b^2\) is even, that doesn't necessarily mean \(a-3\) is even, because the product of an even and any number is even. So we can't deduce anything about \(a\). However, we have shown that if \((a-3) b^2\) is even, then \(b\) must be even, which proves the reverse implication.

Key concepts

Even and Odd Integers

Understanding the classification of integers into evens and odds is fundamental in many areas of mathematics. An even integer is any integer that can be divided by two without leaving a remainder. In mathematical terms, an integer n is even if there exists another integer k such that n = 2k. On the other hand, an odd integer cannot be divided by two evenly and has a remainder of one. Formally, an integer n is odd if there exists an integer k such that n = 2k + 1.

These definitions play a pivotal role when working with integer properties and arithmetic operations. For example, the sum of two even numbers is always even, while the sum of two odd numbers is even as well. Conversely, the sum of an even and an odd number is odd. Similarly, the product of any integer with an even number will result in an even number, which is a key idea in our original exercise regarding the expression \( (a-3) b^2 \) .

Proof Techniques

In mathematics, proof techniques are the methods used to establish the truth of a statement. There are various ways proofs can be conducted:

• Direct Proof: The most straightforward method, typically involving direct application of definitions and properties.
• Indirect Proof or Proof by Contradiction: Involves assuming the opposite of the statement and demonstrating a contradiction.
• Proof by Contrapositive: Proves the negation of a statement as true, thus confirming the statement itself.
• Proof by Induction: Validates a statement for an infinite number of cases by proving a base case and a so-called induction step.

The choice of proof technique depends on the nature of the statement to be proved, the known information, and occasionally the preference of the mathematician. Improving our understanding of these techniques can enhance our ability to tackle a wide range of mathematical problems, including but not limited to, problems dealing with even and odd properties of integers.

Direct Proof

When we choose to use a direct proof, we are pursuing a linear logical path from known truths, like definitions and theorems, to the statement we need to verify. This technique allows us to work directly from the hypothesis of the proposition to the conclusion.

For example, in the provided exercise, we need to show that \( (a-3) b^2 \) is even if and only if \( a \) is odd or \( b \) is even. In the forward implication of our exercise, by knowing the definition of even and odd integers, we directly apply these to the values of \( a \) and \( b \) to establish the evenness of the expression \( (a-3) b^2 \). This step-by-step approach, which builds upon previous knowledge and moves toward the conclusion, encapsulates the essence of direct proof.

Mathematical Reasoning

The exercise provided not only incorporates arithmetic operations with even and odd integers but also requires mathematical reasoning—the process of drawing logical conclusions from given premises. In both the forward and reverse implications of this exercise, we deduce information and infer properties about the integers involved, based on the assumption that \( (a-3) b^2 \) is even.

Mathematical reasoning is what allows us to navigate through the implications of given facts. It helps us to assess whether a statement must be true, might be true, or is definitely not true. For instance, in our exercise, we reason that since the product of an even and any integer is even, then if \( (a-3) b^2 \) is even, the evenness of \( b \) is required, irrespective of \( a \'s \) value. Establishing this process robustly helps students not only to solve problems effectively but also to develop a deeper appreciation for the logic and structure underpinning mathematics.