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 \(x \in \mathbb{R}\). If \(x^{3}-x>0\) then \(x>-1\)

Short answer

The contrapositive of the given statement, 'If \(x\leq -1\) then \(x^{3}-x\leq 0\)', is proven to be true. Therefore, the original statement, 'If \(x^{3}-x>0\) then \(x>-1\)', is also true.

Step-by-step solution

Formulate the Contrapositive Statement

The contrapositive of the statement 'If \(x^{3}-x>0\) then \(x>-1\)' is 'If \(x\leq -1\) then \(x^{3}-x\leq 0\)'. Now, we will prove this contrapositive statement.

Assume the Hypothesis of the Contrapositive Statement

Let's assume that \(x\leq -1\) is true. This is now our hypothesis for the contrapositive statement.

Derive the Conclusion of the Contrapositive Statement

Under the assumption that \(x\leq -1\), we need to prove \(x^{3}-x\leq 0\). We can rewrite this inequality as \(x^{3}\leq x\). Since \(x\leq -1\), both sides of the inequality will be non-positive. Furthermore, since every real number cubed is less than or equal to the number itself when the number is less than or equal to -1, \(x^{3}\leq x\) holds true for all \(x\leq -1\). This proves the contrapositive statement.

Key concepts

Mathematical Logic

Mathematical logic is a subfield of mathematics exploring the applications of formal logic to mathematics. It deals with the concepts of truth, proof and the structure of mathematical arguments. A fundamental part of mathematical logic is understanding how to construct a valid argument to prove the correctness of a statement. In the realm of mathematical proofs, we often use logic to show that if a certain statement is true, then some other related statement must also be true. This involves employing a variety of proof techniques, and one such technique is the contrapositive proof, which is particularly useful when direct proof is challenging.

The original exercise utilizes mathematical logic to address the given inequality, highlighting how contrapositive proof serves as an extension of logical reasoning in mathematical contexts.

Proof Techniques

Proof techniques are the methods used to demonstrate the truth of mathematical statements. These methods include direct proof, indirect proof, proof by contradiction, proof by contrapositive, and mathematical induction, among others. Each technique is suited to different kinds of problems; understanding which to use often depends on the structure of the problem at hand. The contrapositive proof is chosen for the exercise presumably because a direct proof may not be straightforward. This technique involves proving the contrapositive of the given statement: if the result is not true, then the original condition cannot be true either. Use of contrapositive is a powerful technique, especially in dealing with implications and inequalities.

Direct Proof

Direct proof is one of the most common forms of proof where the aim is to start with the given assumptions and use logical steps to arrive at the conclusion. It moves in a straightforward manner from premises to result, usually employing a series of logical implications. While the exercise mentions considering a direct proof, it may not be the simplest approach when dealing with inequalities that involve variables raised to powers, as the relationship between the expressions can become complex. However, understanding direct proof is crucial, as it is frequently the foundation for more complex proof techniques, including the contrapositive method used in this exercise.

Inequalities

Inequalities are mathematical expressions that show the relative size or order of two values. They are fundamental to solving a wide range of problems in mathematics. Knowing how to manipulate and solve inequalities is essential, especially when working with variable expressions. The exercise features an inequality, \(x^3 - x > 0\), which defines the conditions of x that satisfy this particular statement. Inequalities can sometimes be more complex to manage in proofs since they involve a range of values rather than a specific equality. In such cases, other proof techniques like contrapositive can simplify the process as it avoids dealing with the original inequality directly and instead works with its negation or converse.