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Problem Set 1 Real Analysis 1. Let A and B be sets. Prove the following: (a) A∪B=AifandonlyifB⊆A. (note: There are two “directions to prove” in an “if and only if” statement!) (b) A∩B=AifandonlyifA⊆B. (c) A\B=AifandonlyifA∩B=∅. (d) A\B=∅ifandonlyifA⊆B. 2. Let X and Y be sets and A ⊆ X and B ⊆ Y be subsets. For a given function f : X → Y , define the image of A under f to be the set f(A) := {f(x) : x ∈ A} ⊆ Y . Define the preimage of B under f to be the set f−1(B) = {x : f(x) ∈ B} ⊆ X. 3. (a) Prove that f(f−1(B)) ⊆ B. (hint: Let x ∈ f(f−1(B)) be an arbitrary element. Unpack what this means to show that x ∈ B.) (b) Give an example where f(f−1(B)) ̸= B. (c) Prove that A ⊆ f−1(f(A)). (d) Given an example where A ≠ f−1(f(A)). √ √ (a) Prove that 2 is irrational. 3 is irrational. Your argument will be similar to the one given in Lecture 1.2 that (b) Try to adapt your argument from (a) in order to show Explain what goes wrong. 4 is irrational (which is not true). (c) By the same argument as in (a), one can show that 5 is irrational. Prove that 3 + 5 is √√ rational if and only if 3 − 5 is rational; use this and the above to conclude that they both must be irrational. (hint: Consider the product of 3 + 5 and 3 − 5.) √√√√ 4. Prove that the multiplicative identity in a field is unique. (hint: Here’s the setup for the proof. Let F be a field and suppose that a ∈ F and b ∈ F satisfy a·x=x for all x and b·x=x for all x; show that a=b.) 1 √ √√√ 5. Let F be an ordered field. Recall that the positive elements of F are a nonempty subset P ⊆ F satisfying: (i) Ifa,b∈P,thena+b∈P anda·b∈P. (ii) Ifa∈Fanda̸=0,theneithera∈P or−a∈P,butnotboth. (a) Give an example of a nonempty subset P1 ⊆ R that satisfies (i) but not (ii). (b) Give an example of a nonempty subset P2 ⊆ R that satisfies (ii) but not (i). 6. Prove the Transitivity property of inequalities: If x < y and y < z, then x < z. (note: This result may seem “obvious”. If you can’t figure out what to write, reference the definition of x < y from Lecture 1.2 and include this definition in your proof.) 7. LetFbeanorderedfieldandleta,b∈F. (a) Provethata≤bifandonlyifforallε>0wehavea<b+ε. (b) Use(a)toshowthata=bifandonlyifforallε>0wehave|a−b|<ε. 8. (a) Complete the proof of Corollary 1.12 in the notes, which we started in Lecture 1.3. (note: If you’re confused exactly what remains to be shown, just prove the corollary in its entirety.) (b) Complete the proof of the second bullet point of Corollary 1.13 in the notes, which we started in Lecture 1.3. 9. Suppose that A ⊆ B and that both A and B are bounded from above. Prove that sup(A) ≤ sup(B). 10. Is N complete? Justify your answer with a proof or a counterexample. 11. Suppose A is a nonempty set with finitely many elements. Complete the following steps to show using a proof by induction that A admits a maximal element M ∈ A satisfying x ≤ M for all x ∈ A: (a) First prove the base case: if A contains only one element, then A admits a maximal element. This step is easy! (b) Next prove the inductive case: let n ∈ N and assume any set with n elements admits a maximal element. Prove that a set with n + 1 elements also admits a maximal element. (c) Explain why the work you did in (a) and (b) proves the claim. 12. Prove the infimum case (i.e., the second bullet point) of Theorem 1.21 in the notes. 􏰀n􏰁 13.SetA= n+1:n∈N . (a) Prove that sup(A) = 1 using the analytic description of a supremum given by Theorem 1.21 in the notes. (b) Prove that inf(A) = 1/2 using the analytic description of an infimum given by Theorem 1.21 in the notes. 14. Let A ⊆ R be nonempty and bounded from below. Define −A := {−x : x ∈ A}. Prove that −A is nonempty and bounded from above, and moreover that sup(−A) = − inf(A). 2

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