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Chapter 9: Problem 66

We know that if \(A \cdot B=0,\) then either \(A=0\) or \(B=0 .\) If \(A \cdot B=6,\) does that imply that either \(A=6\) or \(B=6\) ? Explain your answer.

Short Answer

Expert verified
Provide a reason for your answer. Answer: The statement is false. There are valid counterexamples, such as \((A, B) = (2, 3)\), where \(A \cdot B = 6\), but neither \(A\) nor \(B\) is equal to 6.

Step by step solution

01

Investigate possible counterexamples

We will try to find pairs of numbers \((A,B)\) such that \(A \cdot B = 6\), but neither \(A=6\) nor \(B=6\). If we can find such a pair, we can say that the statement is not true.
02

Find the pairs of numbers that multiply to 6

Let's find all the possible pairs of integers \((A, B)\) that multiply to get 6 as the result: 1. \(2 \cdot 3 = 6\) 2. \(3 \cdot 2 = 6\) 3. \((-2) \cdot (-3) = 6\) 4. \((-3) \cdot (-2) = 6\)
03

Check if any of the pairs is a counterexample

Looking at the list from Step 2, the pair \((A, B) = (2,3)\) satisfies \(A \cdot B = 6\) without \(A=6\) or \(B=6\). We can also see that both \((-2,-3)\) and \((-3,-2)\) are counterexamples as well.
04

Conclude the result

Based on the counterexamples found in Step 3, we can conclude that the statement "If \(A \cdot B = 6\), then either \(A = 6\) or \(B = 6\)" is not true. There are multiple pairs of numbers \((A, B)\) where \(A \cdot B = 6\) and neither \(A\) nor \(B\) is equal to 6.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Zero Product Property
The Zero Product Property is a fundamental concept in algebra. It states that if the product of two numbers, say \(A\) and \(B\), equals zero, then at least one of the numbers must be zero. This can be summarized as: if \(A \cdot B = 0\), then \(A = 0\) or \(B = 0\) (or both).
This property is extremely useful for solving quadratic equations and other polynomial equations. By setting the equation to zero, you can find the roots by solving for when each factor is zero. Here are some important points to remember about the Zero Product Property:
  • It applies strictly when the product is zero.
  • It does not apply to products resulting in non-zero values.
  • It is a basic but powerful tool in algebraic reasoning.
Counterexample
A counterexample in mathematics is a specific example or case that disproves a general statement or hypothesis. In the context of the given exercise, a counterexample is used to show that the statement "If \(A \cdot B = 6\), then either \(A = 6\) or \(B = 6\)" is incorrect.
To find a counterexample, you look for a scenario where \(A \cdot B = 6\) but neither \(A\) nor \(B\) equals 6. In the step-by-step solution, pairs like (2, 3) and (-2, -3) serve as counterexamples:
  • Both numbers multiply to six.
  • Neither number is six.

Counterexamples are important because they provide concrete proof that a general statement is false, allowing mathematicians to refine or discard incorrect hypotheses.
Integer Multiplication
Integer multiplication involves calculating the product of whole numbers, which can be positive, negative, or zero. In the problem at hand, we are seeking pairs of integers \(A\) and \(B\) that multiply to a certain product, namely 6. This involves:
  • Carefully selecting pairs that achieve the product of 6.
  • Exploring different combinations, including negative numbers, since \((-2) \cdot (-3)\) also results in a positive product of 6.
Integer multiplication is straightforward but becomes richer when considering properties like associativity and commutativity, which allow reordering and regrouping of numbers without affecting the product. These properties help in simplifying complex algebraic expressions and proving mathematical statements.
Algebraic Reasoning
Algebraic reasoning is a key skill in mathematics used to understand and manipulate algebraic expressions and equations. In this exercise, algebraic reasoning involves examining the potential pairs that meet the given condition \(A \cdot B = 6\).

Through algebraic reasoning, students learn to:
  • Systematically identify whether a statement is true or not.
  • Use mathematical properties like the Zero Product Property to solve equations.
  • Provide counterexamples to test the validity of hypotheses.
Algebraic reasoning is not just about crunching numbers. It involves a deeper understanding of why operations work and how different mathematical properties interact with each other, allowing for effective problem-solving and innovation in mathematics.

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Most popular questions from this chapter

A Norman window is composed of a rectangle surmounted by a semicircle whose diameter is equal to the width of the rectangle. (a) What is the area of a Norman window in which the rectangle is \(l\) feet long and \(w\) feet wide? (b) Find the dimensions of a Norman window with area \(20 \mathrm{ft}^{2}\) and with rectangle twice as long as it is wide.

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