Proof of a Lemma regarding absolute values

In summary, the conversation discusses a proof involving the case where |r| = r and the question of why r ≤ |r| holds in this case. It is explained that this is due to the "generalisation" rule of inference and that taking q = |r| would result in a false statement. The lemma stating that |r| ≤ |r| is also mentioned.
  • #1
WWCY
479
12
Hi all,

There's this proof that I've been trying to wrap my head around but it just doesn't seem to sink in. I've attached a screenshot below. Many thanks in advance!

Screen Shot 2018-10-03 at 6.55.13 PM.png


Consider Case 1. There is a step that goes
$$\text{Then} \ |r| = r$$
$$Then -|r| \leq |r| \ \text{and} \ r \leq |r|$$
Why is this the case? This seems to imply that because ##|r|=r##, then ##r \leq |r|##. Is this because of the "generalisation" rule of inference that goes
$$p$$
$$\text{Therefore} \ p \vee q$$
Where ##p = |r| = r## and ##q = |r| > r##? If so, why not write ##q = |r| < r## and get a completely different result altogether?
 

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  • Screen Shot 2018-10-03 at 6.55.13 PM.png
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  • #2
The lemma says ##|r|\leq |r| ## which should be trivial.

For case 1 they already establish ##r=|r| ## so ##r\leq |r|## holds trivially. Yes, ##p\rightarrow p\lor q ## does the trick here.

You may take ##q=|r| ## and then write ##r=|r| \rightarrow r\leq |r| \lor |r|<r ##, but ##q: |r|<r## is simply not true.
 

1. What is the purpose of a lemma in a proof?

A lemma is a smaller, intermediate result that is used to prove a larger theorem. It helps break down complex proofs into manageable steps and supports the overall argument.

2. How are absolute values used in mathematical proofs?

Absolute values are often used in proofs to show the magnitude or distance between two numbers. They can also be used to prove inequalities and to simplify equations.

3. Can a lemma be used to prove multiple theorems?

Yes, a lemma can be used in multiple proofs as long as it is applicable to the specific theorem being proved. Lemmas are often used to prove a variety of theorems in the same field of mathematics.

4. Are there different types of lemmas?

Yes, there are different types of lemmas depending on their purpose in a proof. Some common types include auxiliary lemmas, constructive lemmas, and critical lemmas.

5. How do you know when to use a lemma in a proof?

Deciding when to use a lemma in a proof requires careful consideration of the specific problem at hand. Generally, lemmas are used to simplify a proof or provide a crucial piece of evidence to support the main argument.

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