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

[Terrell]

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.

 

[haruspex]

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?

 

[Terrell]

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

 

[Terrell]

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]

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.

 

[Zafa Pi]

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.

 

[haruspex]

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.

 

[Zafa Pi]

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.