(e) If each fn has at most a countable number of discontinuities, then f hasat most a countable number of discontinuities.

Q: State whether the following statements are true or false: 1. Let S be a non-empty subset of the set of real numbers R. Ir s is bounded above, then Sups is exist, but need not to be unique in general. 2. If A = (-5,5) and B = (5,10), then Inf(A + B) = 10 and Sup(A + B) = 15. 3. The closed interval [1,2] has no maximal element. 4. The set of natural numbers N of R is unbounded. 5. The set of real numbers R, has Sup(R) = o and Inf(R) = -co. 6 The set S= (x E RỊ x? - 25 s 0} has Max(S) = 5 and Inf(S) = -5 with no minimal %3D element. 7. The set S = {1+nez*} has Max(S) = 2 and Min(S) = 1. 8. Every bounded set of real numbers R has maximal and minimal elements. 9. The properties (M2) and (M2) of the definition of the metric space are state that the distance from any point to another is never negative, and that the distance from a point to itself is zero. 10. There are many metric functions d: M x M -R that can be defined on a non-empty set M.

75% For each of the following two statements, give a counterexample to show that the state- ment is false. (a) Let f: N + N be a function. If f is one-to-one, then f is onto. (b) If ACB and AnC +0, then BnC= 0 -ch alt 立

(a) Suppose a, de N. Define what it means to say that d divides a. (b) Find all natural numbers d such that d | 24 but d | 30. (c) Suppose a, b, d E N and that d | a and d | b. Prove that d | a² + b. (d) Suppose X is a non-empty set of real numbers. Define what is meant by an upper bound for X. If X has an upper bound, define what is meant by a supremum for X. (e) Let -{n-1: EN}. {" X = Find (with proof) the supremum of X.

Given any set of 15 integers, must there be two that have the sameremainder when divided by 12? Prove using the pigeonhole principle. Eitherdescribe the pigeons, the pigeonholes, and how the pigeons get to the pigeonholes, or describe a function by giving its domain, co-domain, and how elements of the domain are related to elements of the co-domain.

7) In recent years, many states have passed laws against smoking in public buildings. The total number of states N that have passed a no smoking in public buildings law, t years after 1985 is given by the function N(t) 50 1+19e-0.4t How many states had passed the law in 1985? A) 1.25 B) 0.25 C) 2.5 D) 0

4) Determine Σ (-1)" n! (x-5)" 1=0 34 the interval of conver gence for

Suppose that a number x is to be selected from the real line R, and let A, B, and C be the eventsrepresented by the following subsets of R, where the notation {x : −−−} denotes the set containingevery point x for which the property presented following the colon is satisfied:A = {x : 1 ≤ x ≤ 5}B = {x : 3 < x ≤ 7}C = {x : x ≤ 0} Describe each of the following events as a set of real numbers:(a) Ac(b) A ∪ B(c) B ∩ C c(d) (A ∪ B) ∩ C Problem 1.2Toss a coin 4 times. Let A denote the event that a head is obtained on the first toss, and let Bdenote the event that a head is obtained on the fourth toss. Is A ∩ B empty? Problem 1.3 Consider rolling a six-sided die once. Let A be the set of outcomes where an odd number comes up.Let B be the set of outcomes where a 1 or a 2 comes up. Calculate the sets on both sides of theequalities (Ac ∩ Bc)c = A ∪ B and (Ac ∪ Bc)c = A ∩ B and verify that the equalities hold. Problem 1.4 We are given that P(A) = 0.55, P(B) = 0.35, and P(A∪B) = 0.75. Determine…

Q11: If X and Y are denumerable sets, then X - Y is countable.

If B is countable and f : P(A) → B is a one to one (i.e., injective) function, then A must be finite. (here, P(A) denotes the power set of A) O true O false