Ordering four real numbers such that some conditions are satisfied
$begingroup$
Suppose I have 4 real numbers in $[0,1]$, all different between each other.
Assume that there exists a way of ordering the four numbers such that:
$$
begin{cases}
w_1>w_2\
w_3=1-w_1\
w_4=1-w_2
end{cases}
$$
where $w_1$ is the first number in the ordered sequence, $w_2$ is the second number in the ordered sequence, $w_3$ is the third number in the ordered sequence, $w_4$ is the fourth number in the ordered sequence.
Is such a way of ordering unique? If yes, could you sketch the proof? In not, could you give a counterexample?
combinatorics permutations combinations
asked Dec 20 '18 at 13:13
STFSTF
461422
$endgroup$
|
show 7 more comments
$begingroup$
Suppose I have 4 real numbers in $[0,1]$, all different between each other.
Assume that there exists a way of ordering the four numbers such that:
$$
begin{cases}
w_1>w_2\
w_3=1-w_1\
w_4=1-w_2
end{cases}
$$
where $w_1$ is the first number in the ordered sequence, $w_2$ is the second number in the ordered sequence, $w_3$ is the third number in the ordered sequence, $w_4$ is the fourth number in the ordered sequence.
Is such a way of ordering unique? If yes, could you sketch the proof? In not, could you give a counterexample?
combinatorics permutations combinations
asked Dec 20 '18 at 13:13
STFSTF
461422
$endgroup$
$begingroup$
What exactly do you mean when you say unique?
$endgroup$
– Ankit Kumar
Dec 20 '18 at 13:31
$begingroup$
can you give such an assignment to the following four numbers? $frac{1}{4},frac{1}{8},frac{1}{16},frac{1}{32}$
$endgroup$
– user408906
Dec 20 '18 at 13:38
$begingroup$
@Suraj: I'm assuming that such an assignment exists. My assignment requires that you can construct two pairs summing up to 1. In your example, this is not possible.
$endgroup$
– STF
Dec 20 '18 at 13:41
2
$begingroup$
What about $0,frac{1}{4},frac{3}{4}, 1$ and $frac{1}{4},1,0,frac{3}{4}$? [I choose to do $w_1<w_2$ and swap $w_3$ and $w_4$ it's the same thing.]
$endgroup$
– ancientmathematician
Dec 20 '18 at 13:52
1
$begingroup$
No. Think about it.
$endgroup$
– ancientmathematician
Dec 20 '18 at 13:59
|
show 7 more comments
$begingroup$
Suppose I have 4 real numbers in $[0,1]$, all different between each other.
Assume that there exists a way of ordering the four numbers such that:
$$
begin{cases}
w_1>w_2\
w_3=1-w_1\
w_4=1-w_2
end{cases}
$$
where $w_1$ is the first number in the ordered sequence, $w_2$ is the second number in the ordered sequence, $w_3$ is the third number in the ordered sequence, $w_4$ is the fourth number in the ordered sequence.
Is such a way of ordering unique? If yes, could you sketch the proof? In not, could you give a counterexample?
combinatorics permutations combinations
asked Dec 20 '18 at 13:13
STFSTF
461422
$endgroup$
Suppose I have 4 real numbers in $[0,1]$, all different between each other.
Assume that there exists a way of ordering the four numbers such that:
$$
begin{cases}
w_1>w_2\
w_3=1-w_1\
w_4=1-w_2
end{cases}
$$
where $w_1$ is the first number in the ordered sequence, $w_2$ is the second number in the ordered sequence, $w_3$ is the third number in the ordered sequence, $w_4$ is the fourth number in the ordered sequence.
Is such a way of ordering unique? If yes, could you sketch the proof? In not, could you give a counterexample?
combinatorics permutations combinations
combinatorics permutations combinations
asked Dec 20 '18 at 13:13
STFSTF
461422
asked Dec 20 '18 at 13:13
STFSTF
461422
edited Dec 20 '18 at 13:53
STF
asked Dec 20 '18 at 13:13
STFSTF
461422
asked Dec 20 '18 at 13:13
STFSTF
461422
asked Dec 20 '18 at 13:13
STFSTF
461422
461422
$begingroup$
What exactly do you mean when you say unique?
$endgroup$
– Ankit Kumar
Dec 20 '18 at 13:31
$begingroup$
can you give such an assignment to the following four numbers? $frac{1}{4},frac{1}{8},frac{1}{16},frac{1}{32}$
$endgroup$
– user408906
Dec 20 '18 at 13:38
$begingroup$
@Suraj: I'm assuming that such an assignment exists. My assignment requires that you can construct two pairs summing up to 1. In your example, this is not possible.
$endgroup$
– STF
Dec 20 '18 at 13:41
2
$begingroup$
What about $0,frac{1}{4},frac{3}{4}, 1$ and $frac{1}{4},1,0,frac{3}{4}$? [I choose to do $w_1<w_2$ and swap $w_3$ and $w_4$ it's the same thing.]
$endgroup$
– ancientmathematician
Dec 20 '18 at 13:52
1
$begingroup$
No. Think about it.
$endgroup$
– ancientmathematician
Dec 20 '18 at 13:59
|
show 7 more comments
$begingroup$
What exactly do you mean when you say unique?
$endgroup$
– Ankit Kumar
Dec 20 '18 at 13:31
$begingroup$
can you give such an assignment to the following four numbers? $frac{1}{4},frac{1}{8},frac{1}{16},frac{1}{32}$
$endgroup$
– user408906
Dec 20 '18 at 13:38
$begingroup$
@Suraj: I'm assuming that such an assignment exists. My assignment requires that you can construct two pairs summing up to 1. In your example, this is not possible.
$endgroup$
– STF
Dec 20 '18 at 13:41
2
$begingroup$
What about $0,frac{1}{4},frac{3}{4}, 1$ and $frac{1}{4},1,0,frac{3}{4}$? [I choose to do $w_1<w_2$ and swap $w_3$ and $w_4$ it's the same thing.]
$endgroup$
– ancientmathematician
Dec 20 '18 at 13:52
1
$begingroup$
No. Think about it.
$endgroup$
– ancientmathematician
Dec 20 '18 at 13:59
$begingroup$
What exactly do you mean when you say unique?
$endgroup$
– Ankit Kumar
Dec 20 '18 at 13:31
$begingroup$
What exactly do you mean when you say unique?
$endgroup$
– Ankit Kumar
Dec 20 '18 at 13:31
$begingroup$
can you give such an assignment to the following four numbers? $frac{1}{4},frac{1}{8},frac{1}{16},frac{1}{32}$
$endgroup$
– user408906
Dec 20 '18 at 13:38
$begingroup$
can you give such an assignment to the following four numbers? $frac{1}{4},frac{1}{8},frac{1}{16},frac{1}{32}$
$endgroup$
– user408906
Dec 20 '18 at 13:38
$begingroup$
@Suraj: I'm assuming that such an assignment exists. My assignment requires that you can construct two pairs summing up to 1. In your example, this is not possible.
$endgroup$
– STF
Dec 20 '18 at 13:41
$begingroup$
@Suraj: I'm assuming that such an assignment exists. My assignment requires that you can construct two pairs summing up to 1. In your example, this is not possible.
$endgroup$
– STF
Dec 20 '18 at 13:41
2
2
$begingroup$
What about $0,frac{1}{4},frac{3}{4}, 1$ and $frac{1}{4},1,0,frac{3}{4}$? [I choose to do $w_1<w_2$ and swap $w_3$ and $w_4$ it's the same thing.]
$endgroup$
– ancientmathematician
Dec 20 '18 at 13:52
$begingroup$
What about $0,frac{1}{4},frac{3}{4}, 1$ and $frac{1}{4},1,0,frac{3}{4}$? [I choose to do $w_1<w_2$ and swap $w_3$ and $w_4$ it's the same thing.]
$endgroup$
– ancientmathematician
Dec 20 '18 at 13:52
1
1
$begingroup$
No. Think about it.
$endgroup$
– ancientmathematician
Dec 20 '18 at 13:59
$begingroup$
No. Think about it.
$endgroup$
– ancientmathematician
Dec 20 '18 at 13:59
|
show 7 more comments
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$begingroup$
What exactly do you mean when you say unique?
$endgroup$
– Ankit Kumar
Dec 20 '18 at 13:31
$begingroup$
can you give such an assignment to the following four numbers? $frac{1}{4},frac{1}{8},frac{1}{16},frac{1}{32}$
$endgroup$
– user408906
Dec 20 '18 at 13:38
$begingroup$
@Suraj: I'm assuming that such an assignment exists. My assignment requires that you can construct two pairs summing up to 1. In your example, this is not possible.
$endgroup$
– STF
Dec 20 '18 at 13:41
2
$begingroup$
What about $0,frac{1}{4},frac{3}{4}, 1$ and $frac{1}{4},1,0,frac{3}{4}$? [I choose to do $w_1<w_2$ and swap $w_3$ and $w_4$ it's the same thing.]
$endgroup$
– ancientmathematician
Dec 20 '18 at 13:52
1
$begingroup$
No. Think about it.
$endgroup$
– ancientmathematician
Dec 20 '18 at 13:59