### function 1: is-subset

is-subset:
description:
is-subset(sp) input-sp
returns |yes> if sp is a "subset" of input-sp, else |no>
where subset means: for all kets in sp, they have value <= the value in input-sp
it generalizes the idea of sets and their subsets, since it allows elements to have corresponding coefficients
so for example: asking if {a,c} is a subset of {a,b,c,d,e}
is equivalent to: is-subset(|a> + |c>) (|a> + |b> + |c> + |d> + |e>)
or more compactly: is-subset(split |ac>) split |abcde>
this extends the notion that sets can be represented by "clean" superpositions
and non-clean superpositions represent fuzzy sets
where "clean" means all coefficients of kets are either 0 or 1
examples:
is-subset(|b>) split |abc>
|yes>
is-subset(split |bd>) split |abc>
|no>
-- note, the order of the elements in the superpositions do not matter:
is-subset(8|c> + 13|d> + 0.2|a>) (0.3|a> + 2|b> + 9.7|c> + 13|d>)
|yes>
friends |Fred> => |Sam> + |Emma> + |Rob> + |Liz> + |Harold> + |Tom>
friends |Sam> => |Emma> + |Tom> + |Liz>
friends |Harold> => |Rob> + |Liz> + |Tom> + |Beth>
-- are Sam's friends a subset of Fred's friends?
is-subset(friends |Sam>) friends |Fred>
|yes>
-- are Harold's friends a subset of Fred's friends?
is-subset(friends |Harold>) friends |Fred>
|no>
see also:
is-mbr, union, intersection, superposition

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