"the" mapping? there is no "the" mapping.
you are talking about the canonical inclusion mapping 1 in N to 1 in Z (restriction of the canonical inclusion of rings of integers Z into any other ring, Z is an initial object), which can be seen as a non-generic canonical mapping of semigroups.
but as sets, there is no inherent structure, there are injection, surjections, and of course bijections in both directions.
the only way one can call one set "bigger" is in the very strict sense of sets, N being a true subset of Q. however, this assumes N to be an actual subset of Q, which is a matter of definition and construction. so we say there is some embedding included, which is the same as (re)defining N as that embedded subset, so we are at your canonical inclusion of semigroups again. if you view this as inherent to N and Q, then there are "more" elements in Q as in N, but not in terms of cardinality.
was für eine abartige rechtschreibung von "zuhausi" 🤮
sag bloß, du schreibst mehrzahl "zuhausies" 🤮🤮🤮 angelsächsisch 🤮🤮🤮
(bitte diesen kommentar nicht komplett ernst nehmen)