Question

Prove the following statements with contrapositive proof. (In each case, think about how a direct proof would work. In most cases contrapositive is easier.) Suppose \(a, b \in \mathbb{Z}\). If both \(a b\) and \(a+b\) are even, then both \(a\) and \(b\) are even.

Short answer

According to the contrapositive proof, if either \(a\) or \(b\) is not even, then either \(a b\) or \(a+b\) is not even. This confirms that the original proposition 'If both \(a b\) and \(a+b\) are even, then both \(a\) and \(b\) are even' is true.

Step-by-step solution

Identifying the Contrapositive

Before starting the proof, it's important to correctly identify the contrapositive of the given statement. The contrapositive of 'If both \(a b\) and \(a+b\) are even, then both a and b are even' is 'If either \(a\) or \(b\) is not even, then either \(a b\) or \(a+b\) is not even.' We will now proceed with the contrapositive proof.

Assume either \(a\) or \(b\) is not even

Assuming the negation of the conclusion for the contrapositive proof, let's suppose either \(a\) or \(b\) is not even. This means there exists an odd number among \(a\) and \(b\). Let \(a = 2m + x\) and \(b = 2n + y\), where \(x,y\) belong to \{0,1\} and \(x + y = 1\). This ensures one and only one of \(a\) and \(b\) is odd.

Prove either \(a b\) or \(a+b\) is not even

Based upon assumption, calculate \(a b\) and \(a + b\). \(a b = (2m + x)(2n + y) = 4mn + 2my + 2nx + xy\). Since the first three terms of the expansion are definitely multiples of 2, if \(xy\) is 1 (which is odd), \(a b\) will be odd, hence not even. For \(a + b = (2m + x) + (2n + y) = 2m + 2n + x + y = 2(m + n) + (x + y)\), the first term is multiple of 2. If \(x + y = 1\) (which is odd), \(a + b\) is not even. We have proved the contrapositive.

Key concepts

Direct Proof

A direct proof is one of the most straightforward approaches to validating a mathematical statement. In contrast to proof techniques that may use indirect reasoning, a direct proof lays out a logical sequence of statements that stem from known truths, such as definitions, axioms, or previously proven theorems, to arrive at the given assertion. To perform a direct proof, especially in the context of even and odd integers, you would typically assume that the initial premises about the numbers are true and then use this information to directly demonstrate the truth of the conclusion.

For example, to directly prove that the sum of two even integers is even, one would start by letting the two integers be represented as 2k and 2m (where k and m are integers, since an even number is defined as twice an integer). Adding these gives 2k + 2m, which can be factored to 2(k+m), clearly showing that the sum is an even number since it's twice the integer (k+m).

Mathematical Proof

A mathematical proof is a logical argument that establishes the validity of a mathematical statement beyond a shadow of a doubt. Using a series of rigorously deduced conclusions that follow from basic principles, a mathematical proof aims to demonstrate that a proposition is true in every possible case. There are several proof techniques used in mathematics, such as direct proof, indirect proof (which includes contrapositive and proof by contradiction), and proof by induction.

Proofs are essential in mathematics because they give us confidence in our mathematical reasoning and allow for the building of more complex knowledge on the foundation of solidly established facts. They must adhere to strict logical standards - every step must be justified either by a definition, a postulate, or a previously proven statement. Mathematics values proofs as the final seal of verification that removes any doubt about a proposition's validity.

Even and Odd Integers

Understanding even and odd integers is crucial when dealing with integer properties and in many proof techniques. An even integer is defined as any integer that can be expressed as twice another integer, written in the form 2n where 'n' is an integer. Odd integers, on the other hand, are numbers that are one more or one less than an even integer, and can be represented as 2n + 1 or 2n - 1.

This distinction is important in various mathematical contexts, including proving statements about integer properties. Even and odd integers have different behaviors under addition, subtraction, and multiplication, which can be observed in statements like 'the sum of two even integers is even', or 'the product of an odd and an even integer is even'. These properties are often the foundation upon which more complex proofs are built.

Proof Techniques

Proof techniques are the methods and strategies used to establish the truth of mathematical statements. Familiarizing oneself with various proof techniques is a huge part of becoming proficient in mathematical reasoning. Some commonly used techniques include:

• Direct Proof: As mentioned earlier, it involves a straightforward approach of showing that a conclusion follows logically from the premises.
• Contrapositive Proof: This techinque entails proving the contrapositive of the implication. For a statement 'If P, then Q', the contrapositive is 'If not Q, then not P', which is logically equivalent to the initial statement.
• Proof by Contradiction: This strategy assumes that the statement to be proved is false and shows that this assumption leads to a contradiction, thus the original statement must be true.
• Proof by Induction: Useful particularly for proving statements about integers, it shows that if a statement holds for an initial integer and if true for an integer 'k', it is true for 'k+1'. This bootstrapping principle confirms the statement for all integers above the initial one.

Understanding how to effectively employ different proof techniques can make tackling complex mathematical problems more manageable. For instance, in the exercise provided, a contrapositive proof proved more intuitive than a direct proof because of the ease of working with the negated form of the statement.