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 \(x, y, z \in \mathbb{Z}\) and \(x \neq 0 .\) If \(x \nmid y z,\) then \(x \nmid y\) and \(x \nmid z\)

Short answer

The proof shows that if \(x\) divides either \(y\) or \(z\) then \(x\) divides the product of \(y\) and \(z\), \(yz\). Which is the contrapositive of the given statement and hence proves it.

Step-by-step solution

- Define variables

Define \(x, y, z\) as integers and \(x\) is not equal to 0 that is \(x\neq 0\)

- Contrapositive

The contrapositive of the statement 'If \(x \nmid y z,\) then \(x \nmid y\) and \(x \nmid z\)' is 'If \(x\) divides \(y\) or \(x\) divides \(z\) (alternatively, if it's not true that \(x\nmid y\) and \(x\nmid z\)), then \(x\) divides \(y z\)'

- Prove contrapositive

Assume \(x\) divides \(y\) or \(x\) divides \(z\). In the first case, where \(x\) divides \(y\), there exists an integer \(m\) such that \(y = m*x\). Hence \(y * z = m * x * z\), and so \(x\) divides \(y*z\). In the second case, where \(x\) divides \(z\), there exists an integer \(n\) such that \(z = n*x\). Hence \(y * z = y * n * x\), and so \(x\) divides \(y*z\). In both cases, we proved that if \(x\) divides \(y\) or \(x\) divides \(z\), then \(x\) divides \(yz\) . So the contrapositive is true, which proves the original statement

Key concepts

Mathematical Proof

In mathematics, a proof is an irrefutable demonstration that a certain statement follows from a set of agreed-upon axioms and previously established statements. A mathematical proof must adhere to the rules of logical reasoning and the context of mathematical rigor. It's like putting together the pieces of a puzzle where each piece is a logical argument that fits with the others to reveal an undisputed truth.

Proofs can take various forms. Direct proofs tackle the statement head-on by linking the statement back to axioms or previously proven theorems through a logical sequence of statements. Contrapositive proofs, on the other hand, prove an equivalent statement that translates the original implication's converse into its negative form—which often simplifies the proof process. This technique can be particularly useful in number theory, where properties of integers are frequently explored.

Number Theory

Number theory is a branch of mathematics that is concerned with the properties and relationships of numbers, especially the integers. It has applications in various fields, including cryptography, computer science, and even in solving seemingly simple yet intricate statements involving divisibility, primes, and modular arithmetic. For example, understanding the divisibility rules for integers is essential in number theory.

In the context of the given problem, examining the divisibility of products and understanding how divisors relate to each component of the product are key aspects of what makes number theory both fascinating and complex. As such, number theorists often rely on proof techniques like the contrapositive to explore and validate theorems related to numbers.

Proof Technique

Proof techniques are the various methods used to establish the truth of mathematical statements. Each technique has its utility, depending on the nature of the problem and what is known at the outset. For instance:

• Direct proof: Assumes the premises are true to show the conclusion follows.
• Indirect proof (including contradiction and contrapositive): Assumes some negation of the conclusion in order to show a contradiction arises, meaning the original statement must be true.
• Mathematical induction: Proves a base case and then shows that if the statement holds for an arbitrary case, it also holds for the next case.

Contrapositive proof is particularly powerful as it oftentimes simplifies the burden of proof, especially when the direct route seems intricate or less intuitive. By proving the contrapositive of the original statement, you validate the original implication since they are logically equivalent.

Integers

Integers, denoted as \(\textbf{Z}\), are the set of whole numbers that can be positive, negative, or zero. They form the backbone of many mathematical constructs and are integral to number theory. Fundamental operations like addition, subtraction, multiplication, and division (excluding division by zero) are defined within the set of integers.

When dealing with proofs in number theory, properties of integers such as divisibility play a crucial role. Understanding how integers interact through these operations is essential. The given problem examines a scenario concerning divisibility among integers, a common investigation in number theory, and highlights the importance of integers in creating a foundation for higher mathematical concepts.