Question #141631

A set S is cardinally majorizable by a set T iff there exists a(n) ______________ from T to S.

It is not injection. Bijection from T to S? Maybe so that there will be injection from S to T.


1
Expert's answer
2020-11-02T20:30:54-0500

A map f:XYf: X \to Y is called a surjection if for each yYy\in Y there exists xXx\in X such that f(x)=y.f(x)=y.


A set SS is cardinally majorizable by a set TT iff there exists a surjection from TT to SS.


As bonus for the customer note that if f:TSf: T\to S is a surjection, then we can construct an injection g:STg: S\to T in the following way: for each sSs\in S choose exactly one element tf1(s)t\in f^{-1}(s)\ne\emptyset, and define g(s)=tg(s)=t. Since f1(s1)f1(s2)=f^{-1}(s_1)\cap f^{-1}(s_2)=\emptyset for s1s2s_1\ne s_2, gg is an injection.


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