what does this symbol mean in terms of optimization problems?

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$A\setminus B$ is the set of all the objects that are in the set $A$ but not in the set $B$. Therefore, if the notations in the two links are consistent, $j\in N\setminus \{i\}$ should mean whatever $$j\text{ is a }``\text{damaged component and depot'' different from }i$$ means.