Question

Assume that \(a>0, b>0\). Prove each statement. Hint: Each part requires two proofs: one for \(\Rightarrow\) and one for \(\Leftarrow .\) (a) \(a\frac{1}{b}\)

Short answer

Both parts (a) and (b) are true: \(a \frac{1}{b}\).

Step-by-step solution

Prove the Forward Inference for Part (a)

To prove that \[a<b \Rightarrow a^2 < b^2\]:Assume that \(a < b\). When both sides are positive, squaring both sides of the inequality will maintain the inequality. Therefore, \(a^2 < b^2\). This completes the forward proof.

Prove the Reverse Inference for Part (a)

To prove that \[a^2 < b^2 \Rightarrow a < b\]:Assume that \(a^2 < b^2\). Since both \(a\) and \(b\) are positive, we can take square roots of both sides without changing the inequality. Thus, \(a < b\). This completes the reverse proof.

Prove the Forward Inference for Part (b)

To prove that \[a < b \Rightarrow \frac{1}{a} > \frac{1}{b}\]:Assume that \(a < b\). Since \(a, b > 0\), taking reciprocals will reverse the inequality, so \(\frac{1}{a} > \frac{1}{b}\). This completes the forward proof.

Prove the Reverse Inference for Part (b)

To prove that \[\frac{1}{a} > \frac{1}{b} \Rightarrow a < b\]:Assume that \(\frac{1}{a} > \frac{1}{b}\). Multiplying both sides by \(ab\) (which is positive), we get \(b > a\). Therefore, \(a < b\) completes the reverse proof.

Key concepts

Squaring Inequalities

Squaring inequalities is an essential concept in algebra that involves understanding how inequalities behave when both sides of the inequality are squared. Let's explore this further using the example where we are asked to prove that if \(a < b\), then \(a^2 < b^2\), assuming \(a\) and \(b\) are both positive.

When both numbers are squared, the interval between \(a^2\) and \(b^2\) mirrors the interval between \(a\) and \(b\), only squared. When both numbers are positive, the inequality \(a < b\) simply holds true when squared because:

• Squaring both sides of an inequality involving positive numbers maintains the direction of the inequality.
• If \(a^2 < b^2\), this tells us, just as with \(a < b\), that \(a\) is naturally less than \(b\).
• We can take the square root of both sides without changing the inequality because the square root function is increasing for positive inputs.

In summary, understanding squaring inequalities helps with identifying when and how these mathematical transformations can be applied when dealing with real numbers.

Reciprocal Relationships

Reciprocal relationships occur frequently in inequalities and can sometimes reverse the direction of the inequality. This is because taking the reciprocal of numbers between 0 and 1 yields numbers greater than 1, and vice versa. Let's dive into how this concept is applied in inequalities with positive numbers \(a\) and \(b\), specifically proving that \(a < b\) leads to \(\frac{1}{a} > \frac{1}{b}\).

Here's why it works like that:

• When \(a < b\) and both numbers are greater than 0, their reciprocals will follow the inequality \(\frac{1}{a} > \frac{1}{b}\) because reciprocal operations on positive numbers flip the inequality.
• The inverse operation for reciprocals benefits from the notion that multiplying both sides of an inequality by a positive product maintains the inequality's order. This is used in the proof by multiplying both sides of the inequality \(\frac{1}{a} > \frac{1}{b}\) by \(ab\).

Understanding these reciprocal relationships is crucial as they frequently arise in fractions, ratios, and rates, offering insight into why and how inequalities might switch their directions.

Mathematical Proofs

Mathematical proofs are fundamental in verifying statements and ensuring their validity logically. In the context of inequalities, proofs use logical deductions to confirm relationships between quantities. Let's discuss the two types of inferences involved in proofs: forward and reverse inferences.

**Forward Inference**
When one reads \(a < b\), the goal is to prove the implication that \(a^2 < b^2\) or \(\frac{1}{a} > \frac{1}{b}\).

• This often involves direct use of definitions and properties of operations like squaring or taking reciprocals.
• Clearly laying out each step helps track assumptions and results, providing clarity in the transformation from the premise to conclusion.

**Reverse Inference**
This is essentially working backward. For instance, starting with \(a^2 < b^2\) and proving \(a < b\).

• Often seen in proofs involving reversals, like reciprocals, where reversing an operation requires careful treatment of assumptions.
• Used to show that an implication not only goes one way but in both, satisfying "if-and-only-if" conditions.

Building a solid understanding of mathematical proofs enhances logical reasoning and is a skill that extends beyond mathematics into various fields requiring analytical thinking.