Problem 34 - Discrete Mathematics with Applications

Question

Give a counterexample to disprove each statement, where \(P(x)\) denotes an arbitrary predicate. The square of every real number is positive.

Step-by-Step Solution

Answer

The counterexample to the statement "The square of every real number is positive" is when \(x=0\). Since \(0^2=0\), the square of \(0\) is not positive, thus disproving the statement.

Step 1: Identify the claim to disprove.

First, we have to look at the statement in question: "The square of every real number is positive." In other words, for all real numbers \(x\), \(x^2 >0\).

Step 2: Consider special cases.

The initial statement doesn't restrict the type of real numbers we can use to find a counterexample. Therefore, we should explore negative numbers, positive numbers, and zero as potential counterexamples.

Step 3: Test the special cases.

We can start by testing if negative numbers will provide a counterexample. A negative number squared will always be positive, so it doesn't serve as a counterexample. Positive numbers squared are also positive. Finally, we can consider \(0\). The square of \(0\) is \(0^2=0\), which is not a positive number.

Step 4: Present the counterexample

In this case, the counterexample to the original statement is when \(x=0\). Since \(0^2=0\), the square of \(0\) is not positive. This disproves the statement "The square of every real number is positive," as we have found a real number whose square is not positive.

Key Concepts

Understanding Counterexample

In mathematics, a counterexample is an instance or case which disproves a statement or proposition. This concept shows the importance of specificity in mathematics, as the discovery of a single counterexample is enough to overturn a general statement. Take for instance the claim that 'The square of every real number is positive.' This seems plausible at first glance, because squaring a number often increases its value. However, by simply considering the number zero, we find that its square is not positive, but rather, zero itself. This is the quintessential role of a counterexample; it challenges broad assertions and supports precise mathematical understanding.

The Role of Predicate Logic

When disproving mathematical statements, predicate logic can be a powerful tool. Predicate logic expands on propositional logic by considering the truth-value of statements that include variables. When a statement like 'For all real numbers \(x\), \(x^2 > 0\)' is made, predicate logic is used to assess its validity. This kind of statement is known as a universal quantification because it attempts to apply a rule to all members of a certain set—in this case, the set of real numbers. To disprove it, one must find a specific example for \(x\) that renders the predicate \(x^2 > 0\) false. This demonstrates how predicate logic is essential for both formulating and challenging mathematical claims.

Real Numbers and Their Properties

The set of real numbers is a fundamental concept in mathematics, encompassing all the numbers on the continuous number line. This includes both positive and negative integers, fractions, irrational numbers, and zero. The diversity within the real numbers makes it a vast field for testing general statements. For instance, the claim in question involves the squaring of 'every real number'. Since the set of real numbers includes zero, which when squared does not result in a positive number, it showcases an important property of real numbers: not all operations on all real members yield positive results. This emphasizes the need to consider all subset members when making general statements about real numbers.