Math 131AReal AnalysisQuiz 1Instructions: You have 30 minutes to complete the quiz. There are two problems, worth atotal of 30 points. You may not use any books or notes. Write your solutions in the space belowthe questions. If you need more space, use the back of the page. Do not forget to write your nameand UID in the space below.Name:Student ID number:Question Points Score1 152 15Total: 30Problem 1.For each of the following sentences say whether it is true or false and give a one sentencereason explaining why.(a) [4pts.] If Michael can carry a rhinoceros on planet Earth, then Michael is simulta-neously alive and dead.Solution: Michael cannot carry a rhinoceros, so this sentence is true.(b) [5pts.] ∀x ∈ R, x2> 0 =⇒ x > 0.Solution: It is false: its negation∃x ∈ R, x2> 0 and x ≤ 0, i.e. ∃x ∈ R, x ≤ 0 < x2,is true because −1 ≤ 0 < (−1)2.(c) [6pts.] ∃N ∈ R : ∀n ∈ N, n > N =⇒ |1n− 1| <18.Solution: It is false: its negation∀N ∈ R, ∃n ∈ N :n > N and1n− 1≥18is true since we can take n = max{2, dNe + 1}.Problem 2.In any of the following questions, if you are struggling, try to write out the sentence insymbolic quantifiers.(a) [4pts.] Negate the following sentence.Whenever given x ∈ R, we can find a y ∈ R such that x = y2.Solution: There is an x ∈ R with the property that for any y ∈ R, x 6= y2.(b) [5pts.] Suppose X is a set. Negate the following sentence concerning X.There is a function f : N → X with the property that whatever x we choose in X,we can find an n ∈ N with f (n) = x.Solution: Whatever function f : N → X we pick, we can find an x ∈ X withthe property that for all n ∈ N, f (n) 6= x.(c) [6pts.] Suppose f : R → R. Negate the following sentence concerning f.Whenever given  > 0, we can find a δ > 0 with the property that|x − y| < δ implies |f(x) − f(y)| < .Solution: There is an  > 0 so that whatever δ > 0 we try, we can find x, y ∈ Rwith |x − y| < δ and |f (x) − f (y)| ≥