Minimal Abstraction building blocks

\[(a,b)\] A minimal abstraction with corresponds to several related notions.

• one more than one (single) - the number two
• one and also (together with) another one - a minimal aggregate

A Pair is necessarily structually different from a single value. It is a differnt level of abstraction - an aggregate.

Represented as

• a Set of size 2 (unordered)
• a Tuple (position-based ordering)
• a List of size 2 (position-based ordering)
• a Record of 2 elements (symbol-tagged)

\[\cdot \mapsto \cdot\] A mapping from one arbitrary value to another (arity of 2)

Represented as

• an Ordered Pair

a formal definition of a mental concept (a mental category).

• no notion of an order
• no notion of a duplicate (multiple references to the same element)
• the notion of a membership based on the notion of equality
• the notion of a subset (a selection). may be empty (none) or all
• notion of partitions (non-overlapping subsets) based on a certain property (a disjoint union of non-overlapping subsets)

[(a,b)] An association list is a List of ordered pairs (itself in some particular order).

\[A \rightarrow B\] A Set of Arrows from one Set (called domain) to another (called codomain).

A function, as a whole, is usually considered as a Set of Ordered Pairs (values drawn from corresponding Sets).

Is image is a subset of a Cartesian Product (which is a set of all possible Ordered Pairs). This, BTW, is where structural typing kicks in.

\[f (g (x))\] Nesting is the most universal and subtle notion.

It corresponds to use of parenthesis in mathematics for gouping (putting together) and for establishing an explicit order of application (of nested operators).

The notion of an associative operation is based on the notion of nesting.

A set of keys mapped into a set of values.