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单词 ProofThatARelationIsUnionOfFunctionsIfAndOnlyIfAC
释义

proof that a relation is union of functions if and only if AC


Theorem.

A relationMathworldPlanetmath R is the union of a set of functionsMathworldPlanetmath each of whichhas the same domain as R if and only if for each set A ofnonempty sets, there is a choice function on A.

Proof.

Suppose that R is a relation with dom(R)=A and rng(R)B. Let g:A𝒫(B) be givenby aR[{a}]. There is be a choice functionc on g[A]. Let f=cg, and for each paira,bR, let fab send a to b and agreewith f elsewhere. Let F={faba,bR}. Clearly FA×B, so supposeu,vF; then there is a pair a,bR such that u,vfabF. Either u,v=a,b, or v=f(u)=cg(u)=c(R[{u}])R[{u}]. In each case, u,vR.Thus, FR. For each pair a,bR, a,bfabF, so RF. Therefore, R=F.
Suppose that A is set of nonempty sets. Let R={{a}×aaA}. A set x is an element of dom(R) if and only if x{a} for some aA. Thus,dom(R)=A. There is a set F of functions, each of which hasdomain A, such that R=F. Let fF; thendom(f)=A, and for each pair a,f(a)f,a,f(a){a}×a; i.e., f(a)a.Each such f is, thus, a choice function on A.∎

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