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 \(a, b, c \in \mathbb{Z} .\) If \(a\) does not divide \(b c,\) then \(a\) does not divide \(b\).

Short answer

The statement can be proved by contrapositive proof: If \(a\) divides \(b\), then \(a\) does indeed divide \(bc\). Thus, proving the original statement: If \(a\) does not divide \(bc,\) then \(a\) does not divide \(b\).

Step-by-step solution

Understand the Contrapositive Statement

Contrapositive statement for the given conditional is: If \(a\) divides \(b\), then \(a\) divides \(bc\). This is the statement we need to prove.

Assume that \(a\) divides \(b\)

In order to prove the contrapositive, we start by assuming that the first part of it is true i.e., assume that \(a\) divides \(b\). This means there is some integer \(k\) such that \(b = a \cdot k\).

Demonstrate that \(a\) divides \(bc\)

With the assumption that \(b = a \cdot k\), it follows that \(bc = a \cdot k \cdot c\). Thus, \(a\) divides \(bc\), because \(k \cdot c\) is an integer and \(bc\) can be expressed as a multiple of \(a\).

Key concepts

Mathematical Proofs

Mathematical proofs are rigorous arguments that establish the truth of mathematical statements. A proof must derive a conclusion using a series of logical steps, starting from agreed-upon assumptions and previously proven facts, known as axioms and theorems. There are various techniques to conduct proofs, including direct proof, indirect proof (proof by contradiction), and contrapositive proof. In a direct proof, one demonstrates that the premises lead directly to the conclusion. An indirect proof, on the other hand, involves assuming the negation of the conclusion and showing this leads to a contradiction. The contrapositive proof, a valuable tool in a mathematician's arsenal, hinges on the logic that if a statement is true, then so is its contrapositive. The skill of proving statements is not only foundational in advancing mathematical theories but also fosters critical thinking and problem-solving skills.

Divisibility

Divisibility in mathematics is a property of integers. When we say that an integer 'a' divides another integer 'b', denoted as 'a | b', it means that there is an integer 'k' such that 'b = a * k'. Divisibility is fundamental to the study of number theory and has implications for various algorithms and proofs in mathematics. It's important to understand that divisibility implies no remainder; hence 'a' is a factor of 'b'. The concept often requires us to explore fundamental properties such as primes, common factors, and multiples. Recognizing divisibility patterns and applying them can simplify many problems, particularly in algebra and arithmetic.

Integer Properties

Integers are the set of whole numbers and their negatives, including zero. Properties of integers include operations such as addition, subtraction, multiplication, and division (excluding division by zero). In the realm of divisibility, several properties arise, such as closure, commutative, associative, and distributive properties. An understanding of even and odd integers, primes, and composite numbers is also crucial. These properties serve as tools for proving a variety of statements about integers in both simple and complex mathematical contexts.

For example, closure under multiplication means that the product of any two integers will always be an integer. When dealing with proofs, leveraging the known behaviors of integers allows us to predict and verify the outcomes of operations within our statements.

Proof Techniques

Proof techniques in mathematics are diverse strategies used to establish the truth of statements. The choice of an appropriate proof technique can greatly affect the simplicity and elegance of a proof. Popular methods include:

• Direct Proof: Derives the conclusion by a straightforward application of definitions, theorems, and axioms.
• Proof by Contrapositive: Instead of proving the statement 'if p, then q', one proves 'if not q, then not p', which is logically equivalent to the original statement.
• Proof by Contradiction: Assumes the opposite of what you are trying to prove and demonstrates that this assumption leads to a contradiction.
• Inductive Proof: Proves a base case and then shows that if the statement holds for some case 'n', it must also hold for 'n+1'.

Each technique has its own merits and is often deployed based on the nature of the proposition and the information available. For example, when the direct route may not be apparent, a proof by contrapositive might offer a clearer path to the conclusion. Understanding when and how to apply each technique is part of the mathematical learning process and is key to solving proofs effectively.