Question

Prove the following statements using either direct or contrapositive proof. Suppose \(x \in \mathbb{Z} .\) If \(x^{3}-1\) is even, then \(x\) is odd.

Short answer

The statement 'Suppose \(x \in \mathbb{Z}\). If \(x^{3}-1\) is even, then \(x\) is odd.' is proved using contrapositive proof. The proof shows that if x was even, then \(x^{3}-1\) turns out to be odd. Thus, by laws of logic, if \(x^{3}-1\) is even, then \(x\) is odd.

Step-by-step solution

Understanding the Given Statement

The given statement is already in the form 'If P, then Q', where P is 'x cubed minus 1 is even' and Q is 'x is odd'. This is one form of a logical statement that we can try to prove either directly, by showing that if P holds, then Q follows, or by its contrapositive, i.e., if 'not Q' holds , then 'not P' follows. We will opt for the contrapositive proof in this exercise. So, 'not Q' means x is not odd (i.e., x is even), and 'not P' means x cubed minus 1 is not even (or x cubed minus 1 is odd).

Beginning the Contrapositive Proof

Assume that x is even. By the definition of even, it can be written as x = 2k, for some integer k in the domain of integers.

Substituting for the Contrapositive

Substitute x = 2k into x cubed - 1 and simplify. Thus we have \((2k)^{3}-1=8k^{3}-1\).

Understanding the Parity of Expression

Notice that 8k^{3} and 1 are both odd numbers, because 8k^{3} is the product of two even numbers and an odd number which is even, minus 1 gives an odd result, since the difference between an even and odd number is always odd.

Concluding the Proof

Therefore, if x was not odd (or x was even), x cubed minus one resulted in an odd number. Thus, we have succeeded in proving the contrapositive: If 'not Q', then 'not P'. By laws of logic, this means the original statement 'If P, then Q' is also true. Thus, the statement is proved.

Key concepts

Direct Proof

In mathematics, a direct proof is a straightforward method to establish the truth of a given statement. It involves starting with the hypothesis and, through a series of logical steps and accepted principles, arriving at the conclusion. This method is akin to a domino effect where the initial push (the hypothesis) inevitably leads to the fall of the last domino (the conclusion).

For example, to directly prove that the sum of two even integers is even, we might start with two even integers represented by 2m and 2n, where m and n are arbitrary integers. Adding these yields 2m + 2n = 2(m+n), which is clearly in the form of 2k, where k is another integer. Hence, by definition, the sum is even, confirming the statement directly.

Mathematical Logic

At the heart of proofs lies mathematical logic, a formal system of reasoning that allows us to deduce new truths from established ones. In the realm of logic, statements take on truth values and logical connectives link these statements to form compound expressions. The logical form 'If P, then Q' (implication) is one such expression, where P is the hypothesis and Q is the conclusion.

When proving mathematical statements, understanding the nuances of implications is essential. If 'If P, then Q' is true, but P does not happen, there is no effect on the truth of Q. However, if P does happen and Q is not true, we have discovered a fallacy. It is this structure that we rely upon when selecting a proof technique, such as direct or contrapositive proof.

Parity of Integers

The parity of an integer refers to whether it is even or odd. In general,

• An even integer can be expressed as 2k, where k is an integer.
• An odd integer can be expressed as 2k + 1, again where k is an integer.

The behaviour of numbers under operations such as addition, subtraction, and multiplication is largely influenced by their parity. For instance:
- The sum of two even integers is even.
- The sum of an even and an odd integer is odd.
- The product of any integer with an even integer is even.

Understanding the parity of numbers is fundamental in many proofs, including the textbook exercise which relates the parity of a cube of an integer to the integer itself.

Proof Techniques

A variety of proof techniques exist to establish the veracity of mathematical statements. Some of these include:

• Direct proof: Demonstrate that the conclusion follows logically from the hypothesis.
• Indirect proof (Proof by contradiction): Assume that the conclusion is false and derive a contradiction.
• Contrapositive proof: Instead of proving 'If P, then Q', prove the equivalent statement 'If not Q, then not P'.
• Mathematical induction: Prove a base case is true, then show if an arbitrary case is true, the next case must also be true.

In the given exercise, the contrapositive proof is used, where the truth of the statement 'If x³ - 1 is even, then x is odd' is shown by proving its contrapositive: 'If x is not odd (thus even), then x³ - 1 is not even (thus odd)'. Understanding the appropriateness of each technique in different contexts is an essential skill in mathematical problem-solving.