Subset and vertex-complement notation

PDF p. 7: §3.1, opening notation

Let $X$ and $a$ be sets. The singleton $\{a\}$ is the set whose only element is $a$. Write $X+a=X\cup \{a\}$ and $X-a=X\setminus \{a\}$. Thus $X+a$ consists of the elements of $X$ together with $a$, while $X-a$ consists of the elements of $X$ other than $a$; no assumption that $a\in X$ is required.

Let ${\mathrm{\Gamma }}_0$ be the vertex set of the Coxeter diagram $\mathrm{\Gamma }$. For $I\subseteq {\mathrm{\Gamma }}_0$, write $I^c={\mathrm{\Gamma }}_0\setminus I$ for the complement of $I$ relative to ${\mathrm{\Gamma }}_0$, namely the set of vertices of $\mathrm{\Gamma }$ that are not in $I$.

Coxeter-system, roots, length and weak-order conventions

The unordered pair {x,y} and the singleton {x} = {x,x}

The union ⋃ x of a set, and the binary union a ∪ b := ⋃ {a,b}

The difference a ∖ b, the symmetric difference a △ b, and the complement X ∖ a relative to a set X

Subset x ⊆ y, proper subset x ⊊ y, and the separation notation { z ∈ x : φ(z) }

⋃ ∅ = ∅, ⋃ {a} = a, ⋃ {a,b} = a ∪ b, ⋂ {a} = a, and ⋂ {a,b} = a ∩ b

The singleton $\{a\}$ exists and is uniquely determined by its element, by The unordered pair {x,y} and the singleton {x} = {x,x}. Binary union exists for sets by The union ⋃ x of a set, and the binary union a ∪ b := ⋃ {a,b}, and set difference exists for any two sets by The difference a ∖ b, the symmetric difference a △ b, and the complement X ∖ a relative to a set X. Consequently the expressions $X+a$ and $X-a$ are well-defined, including when $a\mathrm{\notin }X$. The same difference construction defines $I^c$ as a set; since $I\subseteq {\mathrm{\Gamma }}_0$ in the sense of Subset x ⊆ y, proper subset x ⊊ y, and the separation notation { z ∈ x : φ(z) }, it is the complement relative to the specified vertex set. The notation ${\mathrm{\Gamma }}_0$ for that vertex set is supplied by Coxeter-system, roots, length and weak-order conventions.

Part (iii) of ⋃ ∅ = ∅, ⋃ {a} = a, ⋃ {a,b} = a ∪ b, ⋂ {a} = a, and ⋂ {a,b} = a ∩ b, applied to $X$ and $\{a\}$, together with the singleton membership rule in The unordered pair {x,y} and the singleton {x} = {x,x}, gives $z\in X+a$ exactly when $z\in X$ or $z=a$. The membership rule in The difference a ∖ b, the symmetric difference a △ b, and the complement X ∖ a relative to a set X similarly gives $z\in X-a$ exactly when $z\in X$ and $z\mathrm{\ne }a$.