Problem 89 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, explain why or give an example to show why it is false. If \(a\) and \(b\) are real numbers and \(a b \neq 0\), then \(a \neq 0\) or \(b \neq 0 .\)

Step-by-Step Solution

Answer

The statement is true. If \(ab \neq 0\), it means that their product is non-zero, which implies that neither \(a\) nor \(b\) can be equal to zero. If any one of them is equal to zero, the product \(ab\) would be zero, contradicting the given condition. Thus, it must be true that both \(a \neq 0\) and \(b \neq 0\), satisfying the given statement: "If \(ab \neq 0\), then \(a \neq 0\) or \(b \neq 0\)."

Step 1: Analyze the conditions of \(ab = 0\)

If \(ab = 0\), it means that either \(a = 0\), or \(b = 0\), or both \(a = 0\) and \(b = 0\). This is because if either \(a\) or \(b\) is equal to zero, their product will be zero.

Step 2: Consider the conditions of \(ab \neq 0\)

Since the given statement is related to the situation when \(ab \neq 0\), let's consider this case. If \(ab \neq 0\), it means that their product is non-zero, which means neither \(a\) nor \(b\) can be equal to zero. If any one of them is equal to zero, the product \(ab\) would be zero, which contradicts the given condition.

Step 3: Determine the truth of the statement

Now that we have examined the conditions of \(ab \neq 0\), we can deduce that in order for the product \(ab\) to be non-zero, it must be true that \(a \neq 0\) and \(b \neq 0\). This fits the given statement: "If \(ab \neq 0\), then \(a \neq 0\) or \(b \neq 0\)." Since the conditions of \(ab \neq 0\) satisfy the given statement, we can conclude that the statement is true.

Step 4: Explain why the statement is true

The statement is true because, under the condition of \(ab \neq 0\), neither \(a\) nor \(b\) can be equal to zero. If either \(a\) or \(b\) was equal to zero, their product would be zero, which contradicts the given condition. Thus, it must be true that both \(a \neq 0\) and \(b \neq 0\), which is equivalent to the given statement, "If \(ab \neq 0\), then \(a \neq 0\) or \(b \neq 0\)."

Key Concepts

Real Numbers

Real numbers are an essential part of mathematics, forming the building blocks of many concepts. They include all the numbers that you can find on the number line. This means all whole numbers, fractions, and decimals are real numbers.
Real numbers can be either positive, negative, or zero, and can be classified as rational or irrational. Rational numbers are numbers that can be expressed as a fraction of two integers, with a non-zero denominator. Irrational numbers, on the other hand, cannot be expressed as a fraction. Famous examples are \( \sqrt{2} \) and \( \pi \).

• Rational numbers: \( \frac{1}{2}, -3, 0, 7 \)
• Irrational numbers: \( \sqrt{3}, \pi, e \)

Knowing the properties of real numbers is crucial, as they help us solve real-world problems through equations and other mathematical processes. In this math problem, we are discussing real numbers \( a \) and \( b \) which are part of the non-zero product \( ab eq 0 \). Understanding how these numbers behave with operations like multiplication helps in proving mathematical statements.

Mathematical Proof

Mathematical proof is a fundamental process in mathematics for demonstrating the truth of a statement. Proofs ensure mathematically that a concept or statement holds under specified conditions. In proving something, we make logical deductions based strictly on established facts and definitions.
There are various types of proofs, including:

• Direct Proof: The direct approach assumes the truth of the given premises and arrives at the conclusion through logical steps.
• Contradiction Proof: Assumes the opposite of what you want to prove, and shows that this leads to a contradiction, thereby proving the original statement.
• Induction Proof: Used mainly for statements involving sequences or series. It involves proving a base case and then an inductive step.

In analyzing our exercise, a direct proof strategy ensures that when \( ab eq 0 \), both \( a eq 0 \) and \( b eq 0 \). This is because if either were zero, then the product \( ab \) would also be zero, contradicting our initial condition. Proving this involves leveraging both the properties of real numbers and logical reasoning.

Conditional Statements

Conditional statements are "if-then" statements that relate two propositions or conditions. They are a staple of mathematical logic.
The general form is "If \( P \), then \( Q \)," where \( P \) is the hypothesis or assumption, and \( Q \) is the conclusion. These statements can be true or false depending on the logical relationship between \( P \) and \( Q \).
Key concepts to understand them include:

• Converse: Swaps the hypothesis and conclusion. From "If \( P \), then \( Q \)," the converse is "If \( Q \), then \( P \)."
• Inverse: Negates both the hypothesis and the conclusion. "If not \( P \), then not \( Q \)."
• Contrapositive: Both swaps and negates the hypothesis and conclusion. It has the same truth value as the original statement. "If not \( Q \), then not \( P \)."

In the exercise, the statement "If \( ab eq 0 \), then \( a eq 0 \) or \( b eq 0 \)" is a conditional statement. In this case, \( ab eq 0 \) is the condition \( P \), and "\( a eq 0 \) or \( b eq 0 \)" is the conclusion \( Q \). The statement was proven to be true using logical analysis, further enriching our understanding of conditional statements in mathematics.