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Problem 1. Let X and Y be non empty sets. Prove that f : X → Y is a bijection ( one to one and onto) if ? a map g : Y → X satisfying f (g(y)) = y for all y ∈ Y and g(f (x)) = x for all x ∈ X .
Problem 2. Let X be a non-empty set. Let ~ be an equivalence relation on X . Given a ∈ X, recall that [a] denotes the equivalence class of a. Prove that given a, b ∈ X we have
[a] ∩ [b] = ? or [a] = [b] .
Problem 3. Prove that surjective maps, equivalence relations, and partitions are all equivalent concepts.
Problem 4. Let S<3 := {p ∈ Q>0 | p2 < 3} and S>3 := {p ∈ Q>0 | p2 > 3}. Show that S<3 contains no largest member and that S>3 contains no smallest member. Do this as follows: Use the “parabola construction” from Lecture 2 to ?nd the q associated to a given p in S<3 ( or in S>3 ) .
Problem 5. Prove that an arbitrary union of open sets (in RN ) is open and that any ?nite intersection of open sets is open. Is the second statement still true if we intersect in?nitely many open sets together?