Question

Use the method of proof by contradiction to prove the following statements. (In each case, you should also think about how a direct or contrapositive proof would work. You will find in most cases that proof by contradiction is easier.) For every positive \(x \in \mathbb{Q},\) there is a positive \(y \in \mathbb{Q}\) for which \(y

Short answer

Every rational number, including \(x \in \mathbb{Q}\), can always have a rational number \(y=\frac{x}{2}\) which is less than \(x\). Therefore the statement is true.

Step-by-step solution

Understand Proof by Contradiction

Proof by contradiction, also known as reductio ad absurdum, is a mathematical method used to prove a statement is true by assuming it is false and then showing that this assumption leads to a contradiction. The principle behind it is that if the assumption leads to a contradiction, then the assumption must be false, and the original statement must be true. Therefore, to apply this method to the given statement, the first step is to assume the opposite of the statement, which is: There exists a positive \(x \in \mathbb{Q}\) which is such that for all positive \(y \in \mathbb{Q}\), \(y\) is not less than \(x\).

Produce a Contradiction

The next step is to produce a contradiction from the assumption. Considering the definition of rational numbers, we know that any rational number can be expressed as the ratio of two integers. Also, if \(y\) cannot be less than \(x\), it implies that we cannot find any positive rational number smaller than \(x\). But, we can always find a positive rational number smaller than \(x\), namely \(\frac{x}{2}\). This is a contradiction since we assumed that it is impossible to find a \(y\) such that \(y<x\).

Conclusion

By producing a contradiction, we have shown that our assumption is false. This implies that our original statement that for every positive \(x \in \mathbb{Q}\), there exists another positive \(y \in \mathbb{Q}\) that is smaller than \(x\) must be true (as per the principle of proof by contradiction).

Key concepts

Rational Numbers

Understanding what rational numbers are is essential when working with mathematical proofs like the one in our exercise. Rational numbers, denoted by the symbol \( \mathbb{Q} \), are numbers that can be expressed as a fraction or ratio of two integers, where the numerator is any integer and the denominator is any non-zero integer. In symbols, a number \( r \) is rational if it can be written in the form \( r = \frac{a}{b} \) where \( a \) and \( b \) are integers, and \( b \eq 0 \).

Rational numbers include the whole numbers, negative integers, and the fractions you're familiar with—essentially any number where you can cleanly write it out as a simple fraction. For example, \( \frac{1}{2} \) and \( -\frac{4}{5} \) are both rational. This is crucial in proofs involving rationals because their properties allow us to conduct operations like division and find other rational numbers, such as halves or thirds, providing us with a suite of potential counterexamples in a proof by contradiction.

Mathematical Proofs

Mathematical proofs are logical arguments which establish the truth of a mathematical statement beyond any doubt. To achieve this, proofs are constructed using a sequence of logical deductions from axioms (statements taken to be true), definitions, and previously established theorems.

There are various methods of proving a theorem, including direct proof, proof by contradiction (reductio ad absurdum), and contrapositive proof, each with its own application and usefulness depending on the scenario. In direct proof, one directly shows that the statement follows from the axioms and theorems. In a contrapositive proof, instead of proving that 'if \( p \) then \( q \) is true, you prove that 'if not \( q \) then not \( p \) is true, which logically gives the same result. Proof by contradiction, our method of interest, differs by starting with the assumption that the statement to be proven is false, and then demonstrating that this assumption inevitably leads to a logical inconsistency, thereby proving that the statement must be true.

Reductio ad Absurdum

The reductio ad absurdum method, also known as proof by contradiction, is particularly powerful when a direct proof or contrapositive is cumbersome or complex to apply. It begins by assuming the negation of the statement to be proven.In the context of our exercise, the negation of 'for every positive \( x \) in \( \mathbb{Q} \) there is a positive \( y \) in \( \mathbb{Q} \) such that \( y