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Will someone help me with this problem. I'm struggling with it.
Define an equivalence relation R on the set P of positive integers bym and n have the same number of prime factors. Is there a function
for each n?
I don't know how to show if this function exists or not. I just learned that a function is a relation such that if the ordered pair (x,y) and (x,y') are elements in R then, y=y'. I don't know how to apply that here. Please help.
Are you by any chance using Herbert Enderton’s text ELEMENTS OF SET THEORY?
If so, this problem appears in it with a nice theorem about function compatible with equivalence relations.
If not, here is the basic idea. Consider the function.
It should be clear to you that ifhave the same number of prime factor then so do
.
In other words,this is enough to say
is compatible with
.
Thus there exists.
If you need details your library should a copy of that text.