I'm having trouble seeing the big picture of this proof.

In summary, the conversation is discussing the proof of (n-r+1, r) being the number of r-combinations of X which contain no consecutive integers. There is a disagreement about whether the establishment of a bijection proves this result or not. After further discussion and thought, it is agreed that the bijection does prove the result, as both the original set and the "new" set have the same number of elements and the same number of ways to choose r-elements. There is a minor disagreement about the language used to describe the sets, but it is concluded that this is a minor issue.
  • #1
Terrell
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I don't see how it proves that (n-r+1, r) is the number of r-combinations of X which contain no consecutive integers.
 

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  • #2
Terrell said:
I don't see how it proves that (n-r+1, r) is the number of r-combinations of X which contain no consecutive integers.
Is it that you do not see that it is a bijection, or that establishment of the bijection proves the result?
 
  • #3
haruspex said:
Is it that you do not see that it is a bijection, or that establishment of the bijection proves the result?
i can't see that the establishment of the bijection proves the result. please do help me
 
  • #4
haruspex said:
Is it that you do not see that it is a bijection, or that establishment of the bijection proves the result?
after giving it some thought, i think I've got it. after applying the bijective function to set S, we can observe that the number of elements of the original set is equal to the number of elements in the set produced by the bijective funtion. however, the only difference is that the original set consists of non-consecutive integers and the "new" set consists of consecutive integers. since both of the sets contains n-r+1 elements, the number of ways to choose r-elements from the "new" set is (n-r+1, r) which should also equal for the original set. did i got that one right?
 
  • #5
Terrell said:
after giving it some thought, i think I've got it. after applying the bijective function to set S, we can observe that the number of elements of the original set is equal to the number of elements in the set produced by the bijective funtion. however, the only difference is that the original set consists of non-consecutive integers and the "new" set consists of consecutive integers. since both of the sets contains n-r+1 elements, the number of ways to choose r-elements from the "new" set is (n-r+1, r) which should also equal for the original set. did i got that one right?
Yes.
 
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  • #6
haruspex said:
Yes.
I think you were being too generous. When Terrell said, "original set consists of non-consecutive integers and the "new" set consists of consecutive integers.", he was referring to S and f(S) and that is in fact not necessarily true.
 
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  • #7
Zafa Pi said:
I think you were being too generous. When Terrell said, "original set consists of non-consecutive integers and the "new" set consists of consecutive integers.", he was referring to S and f(S) and that is in fact not necessarily true.
You are right, I missed that it said "consists of" instead of "may contain", but that is probably just a minor slip in expressing it.
 
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  • #8
haruspex said:
You are right, I missed that it said "consists of" instead of "may contain", but that is probably just a minor slip in expressing it.
Agreed.
 
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1. How can I improve my understanding of the big picture in a proof?

One way to improve your understanding is to break the proof down into smaller, more manageable steps. This will allow you to focus on each individual step and how it contributes to the overall picture. You can also try discussing the proof with a colleague or seeking clarification from the author of the proof.

2. Why is it important to see the big picture in a proof?

Seeing the big picture in a proof allows you to understand the overall logic and structure of the argument. This can help you to identify any flaws or gaps in the proof and also make connections to other concepts and theories.

3. What strategies can I use to visualize the big picture in a proof?

One strategy is to create a visual representation, such as a diagram or flowchart, of the proof. This can help you to see the relationships between different parts of the proof and how they fit together. You can also try explaining the proof to someone else, as this can help you to solidify your understanding.

4. How can I determine what the key points are in a proof?

To determine the key points in a proof, you can start by identifying the main conclusion or theorem being proven. Then, look for any assumptions or definitions that are crucial to the proof. Finally, pay attention to any major steps or lemmas that are used in the proof.

5. What should I do if I still can't see the big picture in a proof?

If you are still struggling to see the big picture in a proof, it may be helpful to take a break and come back to it later with a fresh perspective. You can also try looking at similar proofs or seeking help from a mentor or tutor who may be able to offer a different perspective and explain the proof in a different way.

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