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Is it that you do not see that it is a bijection, or that establishment of the bijection proves the result?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.
i can't see that the establishment of the bijection proves the result. please do help meharuspex 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?haruspex said:Is it that you do not see that it is a bijection, or that establishment of the bijection proves the result?
Yes.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?
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.haruspex said:Yes.
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.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.
Agreed.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.
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.
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.
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.
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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.