Proof by Induction: Explaining Step 3 to 4 | Math Homework

In summary, the conversation discusses a mistake in the notes provided and the correction made in step 3. The question is about the combination of two terms in the inequality and how it relates to Union Bound (Boole's Inequality).
  • #1
Robb
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Homework Statement


Attached are notes from class. Can someone please explain what happens to (-x(n+1)) in step 3 to step 4. Not sure why it goes away. Thanks!

Homework Equations

The Attempt at a Solution

 

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  • #2
Robb said:

Homework Statement


Attached are notes from class. Can someone please explain what happens to (-x(n+1)) in step 3 to step 4. Not sure why it goes away. Thanks!

Homework Equations

The Attempt at a Solution

Your notes (did you take them?) aren't very helpful, as there is a relatively minor error in them -- a misplaced parenthesis.
Here's the corrected version of step 2.
##\Pi_{i = 1}^{n + 1} (1 - x_i) \ge \left(1 - \sum_{i =1}^n x_i)\right)(1 - x_{n + 1})##
The right side of your inequality is incorrect, due to the parentheses being in the wrong place. One of the factors in the inequality is ##(1 - \sum_{i = 1}^n x_i)## and the other is ##(1 - x_{n + 1})##

The error is corrected in step 3. Now, to answer your question, what do ##-\sum_{i = 1}^n x_i## and ##-x_{n + 1}## combine to make?
 
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  • #3
the only things I'd add are

i.) if OP has proven Union Bound (Boole's Inequality), this result follows almost immediately -- though you need to think carefully about sets and coin tossing to get the result. - - - -
edit: geometric mean idea ran into too much trouble. Boole's Inequality still gives a very quick and satisfying answer.
 
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1. What is the purpose of step 3 in proof by induction?

Step 3 in proof by induction is used to show that the statement being proved holds for the next integer after the base case. This step is crucial in establishing the validity of the proof for all integers.

2. How is step 3 different from step 4 in proof by induction?

Step 3 is used to show that the statement holds for the next integer, while step 4 is used to show that the statement holds for all integers. Step 3 is a specific case of step 4, as it only applies to the next integer after the base case.

3. Why is step 3 necessary in proof by induction?

Step 3 is necessary in proof by induction because it bridges the gap between the base case and the general case. It shows that if the statement holds for one integer, it will also hold for the next integer, thus proving the statement for all integers.

4. How do you choose the value for the next integer in step 3?

The value for the next integer in step 3 is chosen based on the statement being proved. It should be the next integer after the base case that will help in proving the statement for all integers. This value is typically chosen by observation and understanding of the problem at hand.

5. Can step 3 be skipped in proof by induction?

No, step 3 cannot be skipped in proof by induction as it is a crucial step in establishing the validity of the proof. Without step 3, there would be a gap in the proof and it would not be considered complete.

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