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Real analysis

Given a function f and a subset A of its domain, let f(A) represent the range of f over the set A; f(A)={f(x) : x belong to A}.
a-Let f(x)=x^2. If A=[0,2](the closed interval{x belong to R : 0<=x<=2}) and B=[1,4],find f(A) and f(B).Does f(A intersection B)=f(A) intersection f(B) in this case?.Does f(A U B)=f(A) U f(B)?.

b-Find two sets A and B for which f(A intersection B) does not = f(A) intersection f(B)

c-show that for an arbitrary function g:R-->R, it is always true that g(A intersection B) subset or equal g(A) intersection g(B) for all sets A,B subset or equal to R

d-Form and prove a conjecture about the relationship between g(A U B) and g(A) U g(B) for an arbitrary function g.

Solution Summary

Topics involved in this solution set include intersections of sets (does f(A U B) = f(A) U f(B)?), and forming a conjecture regarding these intersections.