Say that two natural numbers $i,j$ are separated if $\lvert i-j\rvert>M$, where $M>0$ is a constant. In the paper I'm reading, the author defined $\Gamma$ as "the set of tuples $(i,i',j,j')\in [n]^4$ such that none of the indices $i,i',j,j'$ is separated from the others and $\Gamma^c$ is its complement". Here, $[n]:=\{1,\dotsc,n\}$, and $[n]^4=[n]\times [n]\times[n] \times [n]$.
As far as I understand, I can write $\Gamma=\{A\in[n]^4:\nexists k\in A: \forall l \in A\setminus\{k\}: \lvert k-l\rvert>M \}$.
This definition is not clear to me. Is $\Gamma$ equivalent to :
- "the set of tuples $(i,i',j,j')\in [n]^4$ such that each $i,i',j,j'$ is not separated from the others ", that is, $\Gamma=\{A\in[n]^4:\forall k\in A: \forall l \in A\setminus\{k\}: \lvert k-l\rvert\leq M \}$; or
- "the set of tuples $(i,i',j,j')\in [n]^4$ such that each $i,i',j,j'$ is not separated from some other index ", that is, $\Gamma=\{A\in[n]^4:\forall k\in A: \exists l \in A\setminus\{k\}: \lvert k-l\rvert\leq M \}$.
Can you help me? Additionaly, what would be $\Gamma^c$?
If 1 is true, then $$\Gamma^c=\{A\in[n]^4:\exists k\in A: \exists l \in A\setminus\{k\}: \lvert k-l\rvert> M \}.$$ If 2 is true, then $$\Gamma^c=\{A\in[n]^4:\exists k\in A: \forall l \in A\setminus\{k\}: \lvert k-l\rvert> M \}.$$
I appreciate any help to clarify this. Thanks in advance!
The right answer is 1. And you can even simplify your expression with a little trick. Since $|k - k| \leqslant M$ is always true for $M \geqslant 0$, you get :
$$ \Gamma=\{A\in[n]^4:\forall k\in A, \forall l \in A, \lvert k-l\rvert\leq M \}$$
Hence,
$$ \Gamma^c=\{A\in[n]^4:\exists k\in A, \exists l \in A, \lvert k-l\rvert> M \}$$