Problem 71 - Calculus Early Transcendentals: Pearson New International Edition

Question

Which of the following are true? Assume that \(x\) and \(y\) are real numbers. (a) For every \(x, x>0 \Rightarrow x^{2}>0\). (b) For every \(x, x>0 \Leftrightarrow x^{2}>0\). (c) For every \(x, x^{2}>x\). (d) For every \(x\), there exists a \(y\) such that \(y>x^{2}\). (e) For every positive number \(y\), there exists another positive number \(x\) such that \(0

Step-by-Step Solution

Answer

Statements (a), (d), and (e) are true.

Step 1: Understanding Statement A

This statement claims that for every real number \(x\) that is greater than 0, \(x^2\) is also greater than 0. Since a positive number squared is still positive, this statement is true.

Step 2: Analyzing Statement B

This statement claims an equivalence for all \(x\): \( x>0 \) if and only if \( x^2>0 \). However, \(x^2 > 0\) for any \(x eq 0\), including negative numbers, making this statement false because \( x>0 \) is not the only condition to have \( x^2 > 0 \).

Step 3: Verifying Statement C

This statement suggests that \(x^2 > x\) for all real numbers \(x\). This is not true for \(x=0\) or \(x=1\) as they both make \(x^2=x\). This statement is false.

Step 4: Examining Statement D

This statement suggests that for every real number \(x\), there exists a real number \(y\) such that \(y > x^2\). This holds true because we can always find a number greater than \(x^2\). For instance, \(y = x^2 + 1\) would satisfy the condition. This statement is true.

Step 5: Assessing Statement E

This statement proposes that for every positive number \(y\), there exists another positive number \(x\) such that \(0 < x < y\). This is always true because we can choose \(x = \frac{y}{2}\), which will be positive and less than \(y\).

Key Concepts

Understanding Real Numbers

Real numbers are a fundamental concept in mathematics, including both rational and irrational numbers. They can be thought of as numbers that can represent any point on a number line. This includes:

• Positive numbers (e.g., 5, 3.2)
• Negative numbers (e.g., -2, -7.5)
• Zero (0)
• Fractions (e.g., 1/2, 3/4)
• Irrational numbers (e.g., \(\sqrt{2}, \pi\))

Real numbers are distinguished from complex numbers, which include imaginary numbers. In our exercise, when we say that \(x\) and \(y\) are real numbers, it means we are only considering values that fall on the real number line, excluding any imaginary components.
Understanding real numbers is crucial before delving into problem-solving and inequalities, because it sets the stage for the types of values variables like \(x\) and \(y\) can assume.

Effective Problem Solving Strategies

Effective problem solving in mathematics often involves breaking down the problem into more manageable parts and analyzing these segments one by one. It’s essential to:

• Understand the problem: Make sure you fully grasp what the problem is asking. Break it down into smaller steps if needed, as seen in our example with assessing statements one by one.
• Develop a plan: Decide on the approach or strategies you might use, like using examples, algebraic manipulations, or graphical representations.
• Execute the plan: Carry out your strategy to solve the problem. Always stay organized, trying one method at a time to see which works best.
• Review: Reflect on the solution once you find it to ensure its correctness and consider whether there might be an even more efficient method.

Different problems may require different strategies, so flexibility in thinking and approach is invaluable. In tackling the exercise, this meant deeply evaluating the validity of complex statements such as those in the context of real numbers.

Interpreting Inequality Statements

Inequality statements in mathematics express relationships between numbers, showing how one value compares in size to another. Common inequality symbols include:

• \(>\) - Greater than
• \(<\) - Less than
• \(\geq\) - Greater than or equal to
• \(\leq\) - Less than or equal to
• \(eq\) - Not equal to

In the original exercise, these inequalities were used to establish relationships between \(x\), \(y\), and their functions. For example, statement D introduces an inequality \(y > x^2\), which is true for any \(x\) because one can always find a \(y\) slightly larger than \(x^2\) (such as \(x^2 + 1\)).
Understanding inequalities is vital for solving such statements, especially when interpreting real-world constraints or conditions within mathematical problems.

Crafting a Mathematical Proof

Mathematical proofs are logical arguments that demonstrate the truth or falsehood of a statement using mathematics. Here’s a basic process:

• Define: Clearly state what you want to prove.
• Assumptions: Lay out any initial conditions or assumptions.
• Logical Steps: Develop a chain of reasoning that links your assumptions to the conclusion. This can involve using known theorems or logical deductions.
• Conclusion: Clearly state the result that was proven, tying back to your original statement.

In the exercise, the proof for statements such as E followed these steps by showing that \(x = \frac{y}{2}\) meets the specified conditions. This helps verify the truth of the statement by constructing a logical case with clear assumptions and conclusions.
Learning to write proofs helps in developing critical thinking and logical reasoning skills, which are not only crucial for mathematics but also in making reasoned arguments in everyday life.