Problem 29
Question
Assume that \(a>0, b>0\). Prove each statement. Hint: Each part requires two proofs: one for \(\Rightarrow\) and one for \(\Leftarrow\). (a) \(a\frac{1}{b}\)
Step-by-Step Solution
Verified Answer
(a) True: Both directions of the equivalence are proven true. (b) True: Both directions of the equivalence are proven true.
1Step 1: Prove One Direction of Statement (a)
To prove \(a < b \Rightarrow a^2 < b^2\):Since \(a < b\), subtract \(a\) from both sides: \(b - a > 0\).We know \(b - a > 0\) and \(b + a > 0\) (since both \(a\) and \(b\) are positive).Multiply these inequalities: \((b - a)(b + a) > 0\)This simplifies to \(b^2 - a^2 > 0\).Therefore, \(a^2 < b^2\).
2Step 2: Prove the Other Direction of Statement (a)
To prove \(a^2 < b^2 \Rightarrow a < b\):Since \(a^2 < b^2\), subtract \(a^2\) from both sides: \(b^2 - a^2 > 0\).Factor the difference of squares: \((b - a)(b + a) > 0\).Since both \((b + a)\) is positive, then \((b - a)\) must be positive, meaning \(b - a > 0\).Thus, \(a < b\).
3Step 3: Prove One Direction of Statement (b)
To prove \(a < b \Rightarrow \frac{1}{a} > \frac{1}{b}\):Since \(a < b\), multiply through by \(\frac{1}{ab}\) (this preserves inequality direction as both \(a\) and \(b\) are positive):\(\frac{1}{b} < \frac{1}{a}\).Therefore, \(\frac{1}{a} > \frac{1}{b}\).
4Step 4: Prove the Other Direction of Statement (b)
To prove \(\frac{1}{a} > \frac{1}{b} \Rightarrow a < b\):Assume \(\frac{1}{a} > \frac{1}{b}\). Multiply through by \(ab\) to clear denominators:\(b > a\).Thus, \(a < b\).
Key Concepts
Proof by ContradictionDifference of SquaresReciprocal InequalitiesMathematical Logic
Proof by Contradiction
Proof by contradiction is a powerful technique often used in mathematics to prove statements. It begins by assuming that the statement you want to prove is false. Then, through a series of logical deductions, you attempt to arrive at a contradiction, something that is clearly false or logically impossible. Finding this contradiction implies that the original assumption must be incorrect, and therefore, the initial statement must be true.
For example, consider proving the statement: "If a statement is true, then some condition holds." The proof starts with the opposite assumption; that the statement is true, but the condition does not hold. Through logical reasoning, if this leads to a contradiction, the only conclusion is that the statement must be true if the condition is false.
For example, consider proving the statement: "If a statement is true, then some condition holds." The proof starts with the opposite assumption; that the statement is true, but the condition does not hold. Through logical reasoning, if this leads to a contradiction, the only conclusion is that the statement must be true if the condition is false.
- To use proof by contradiction, follow these general steps:
- Assume the opposite of what you need to prove.
- Use logical reasoning and other known facts to reach a contradiction.
- Conclude the original statement must be true because its negation leads to a contradiction.
Difference of Squares
The difference of squares is a useful algebraic identity that helps simplify the expressions we often encounter. The formula for the difference of squares is: \[ (a^2 - b^2) = (a - b)(a + b) \].
This identity is particularly helpful when solving inequalities, as seen in the statement: if \( a < b \), then \( a^2 < b^2 \). This can be expressed as \( (b - a)(b + a) > 0 \) since both \( a \) and \( b \) are positive, which means \( b - a > 0 \).
This identity is particularly helpful when solving inequalities, as seen in the statement: if \( a < b \), then \( a^2 < b^2 \). This can be expressed as \( (b - a)(b + a) > 0 \) since both \( a \) and \( b \) are positive, which means \( b - a > 0 \).
- Always remember:
- The difference of squares can be transformed into a product of two simpler binomials.
- This transformation is useful for factorization.
Reciprocal Inequalities
Reciprocal inequalities involve comparing the reciprocals of positive numbers, adding a layer of complexity. If \( a < b \) but both are positive, then their reciprocals swap the inequality direction: \( \frac{1}{a} > \frac{1}{b} \).
This is because dividing one unit by a smaller number yields a larger number than dividing it by a larger number. Consider a tablespoon of sugar; if you use smaller cups to distribute it, each cup gets more sugar than if larger cups are used.
This is because dividing one unit by a smaller number yields a larger number than dividing it by a larger number. Consider a tablespoon of sugar; if you use smaller cups to distribute it, each cup gets more sugar than if larger cups are used.
- Key points for understanding reciprocal inequalities:
- The inequality direction reverses with reciprocals of positive numbers.
- This reversal can be remembered by understanding the nature of division and how distribution results change.
- Ensure that the numbers being compared are indeed positive to apply these rules correctly.
Mathematical Logic
Mathematical logic forms the basis for constructing valid mathematical arguments. It incorporates reasoning techniques and principles to reach valid conclusions from given premises. In our exercise, mathematical logic is used to transition between statements like \( a < b \) to \( a^2 < b^2 \) and \( \frac{1}{a} > \frac{1}{b} \).
Logical reasoning helps:
Logical reasoning helps:
- Transition between related statements.
- Utilize known principles, like the difference of squares or reciprocal roles, to solve problems.
- Understand implications of inequalities within a logical framework.
Other exercises in this chapter
Problem 29
In Problems 15-30, specify whether the given function is even, odd, or neither, and then sketch its graph. \(g(t)= \begin{cases}1 & \text { if } t \leq 0 \\ t+1
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In Problems 29-32, show that each equation is an identity. \(\tan \left(\sin ^{-1} x\right)=\frac{x}{\sqrt{1-x^{2}}}\)
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Find the value of each of the following; if undefined, say \(\mathrm{so}\). (a) \(0 \cdot 0\) (b) \(\frac{0}{0}\) (c) \(\frac{0}{17}\) (d) \(\frac{3}{0}\) (e) \
View solution Problem 30
In Problems \(29-34\), find an equation for each line. Then write your answer in the form \(A x+B y+C=0\). Through \((3,4)\) with slope \(-1\)
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