Processing math: 96%

Table and Set Notation

Notation	Equivalent SML with `SET` and `ORDSET`
$\|X\|$	`size` $S$
$\{\}$	`empty` $()$
$\{k\}$	`singleton` $k$
$k \in S$	`find` $X$ $k$
$X \cup \{k\}$	`insert` $(X, k)$
$X \setminus \{k\}$	`delete` $(X, k)$
$\{k \in X\ \|\ e\}$	`filterKey (fn` $k$ `=>` $e$ `)` $X$
$X \cup Y$	`union` $(X, Y)$
$X \cap Y$	`intersection` $(X, Y)$
$X \setminus Y$	`difference` $(X, Y)$

Notation	Equivalent SML with `TABLE` and `ORDTABLE`
$\|T\|$	`size` $T$
$\{\}$	`empty` $()$
$\{k \mapsto v\}$	`singleton` $(k, v)$
$\{k \mapsto e : (k \mapsto v) \in T\}$	`mapKey (fn` $(k, v)$ `=>` $e$ `)` $T$
$\{k \mapsto e : k \in X\}$	`tabulate (fn` $k$ `=>` $e$ `)` $X$
$\{(k \mapsto v) \in T\ \|\ e\}$	`filterKey (fn` $(k, v)$ `=>` $e$ `)` $T$
$\{e_1 : (k \mapsto v) \in T\ \|\ e_2\}$	`mapKey (fn` $(k, v)$ `=>` $e_1$ `) (filterKey (fn` $(k, v)$ `=>` $e_2$ `)` $T$ `)`
$T_1 \cup T_2$	`union (fn (_, v) => v)` $(T_1, T_2)$
$T_1 \cap T_2$	`intersection (fn (_, v) => v)` $(T_1, T_2)$
$T_1 \cap T_2$	`difference` $(T_1, T_2)$
$\sum_{(k \mapsto v) \in T} e$	`reduce add 0 (mapKey (fn` $(k, v)$ `=>` $e$ `)` $T$ `)`

The meaning of add and 0 in the reduce will depend on the type (e.g. Int.+ and 0 for integers or Real.+ and 0.0 for reals). We can also replace $\sum$ with other operations such as $\min$ and $\max$ with the implied semantics.