Let f A rightarrow B be a function Define a relation on A

Let f : A rightarrow B, be, a, function. Define, a, relation ~ on A, as, follows, ~ f = {((x, y) .Given, any, set, .A, and, any, equivalent, relation ~ on, it,, show, that, there, exists, a, set, B, together, with, a, function f : A rightarrow B, such, that,

Solution

Given that f is a funciton from A to B.

The relation ~ is such that

~f = {(x,y) is in AXA f(x) =f(y)}

Hence for any set A with elements x1, x2... say

then f(x1) is in B, f(x2) is in B etc.

Thus for anyt set A and the relation ~ there exists a set B