HW #1 due Friday April 11
[Solow] denotes the book. Since not everybody has it I'm typing in these problems.
Turn in
Ch 1, problems 4-9
4. Unpack these English sentences to determine the hypothesis and the conclusion.
a. The sum of the first n positive integers is n(n+1)/2.
b. r is irrational if r is real and satisfies r2=2.
c. If p and q are positive real numbers with SquareRoot(p*q) not = (p+q)/2, then p not = q.
d. When x is a real number, the minimum value of x(x-1) is at least -1/4.
5. "If I do not get my car fixed, I will miss my job interview," says Jack. Later, you come to know that Jack's car was repaired but that he missed his job interview. Was Jack's statement true or false? Explain.
6. Using Table 1.1 (namely, our class definition of "implies" as a mathematical term), determine the conditions on the hypothesis and the conclusion under which the following statements are true and false and give your reason.
a. If 2>7, then 1>3.
b. If 2<7, then 1<3.
c. If x=3, then 1<2.
d. If x=3, then 1>2.
7. Suppose someone says to you that the following statement is true: "If Jack is younger than his father, then Jack will not lose the contest." Did Jack win the contest?
8. If you are trying to prove that "A implies B" and you know B is false, do you want to show A is true or false? Explain.
9. By considering what happens when A is true and when A is false, it was decided that when trying to prove the statement "A implies B" is true, you can assume that A is true and your goal is to show that B is true. Use the same type of reasoning to derive another approach for proving that "A implies B" is true by considering what happens when B is true and when B is false.
Ch 7, problems 2,7,8.
2. Rewrite each of the following using nested quantifiers.
a. A set S of real numbers has the property that, no matter which element is chosen in the set, you can find another element in the set that is strictly larger.
b. A function of a single variable has the property that for some real number, the absolute
value of the function is always less than that real number
7. Prove that for every real number x>2, there is a real number y<0 such that x=2y/(1+y).
8. Prove that if S = {real numbers x>0 : x2 < 2}, then for every real number epsilon > 0, there is an element x in S such that x2 < 2 - epsilon.
Read Chapter 1 and the other problems within.
I. Let (V,E) be a graph and V' a subset of V, E' a subset of E. What condition need we add in order to say that (V',E') is also a graph? We did this one in class.
(turn in this one too)
II. For p a person and t a time, let fooled(p,t) be true if p is fooled at time t.
Use quantifier notation to write out "You can fool some of the people all of the time, and all of the people some of the time, but you can't fool all the people all of the time." (Abraham Lincoln, paraphrased)
More generally, how is "but" rendered in mathematics-ese?
Turn in
Ch 1, problems 4-9
4. Unpack these English sentences to determine the hypothesis and the conclusion.
a. The sum of the first n positive integers is n(n+1)/2.
b. r is irrational if r is real and satisfies r2=2.
c. If p and q are positive real numbers with SquareRoot(p*q) not = (p+q)/2, then p not = q.
d. When x is a real number, the minimum value of x(x-1) is at least -1/4.
5. "If I do not get my car fixed, I will miss my job interview," says Jack. Later, you come to know that Jack's car was repaired but that he missed his job interview. Was Jack's statement true or false? Explain.
6. Using Table 1.1 (namely, our class definition of "implies" as a mathematical term), determine the conditions on the hypothesis and the conclusion under which the following statements are true and false and give your reason.
a. If 2>7, then 1>3.
b. If 2<7, then 1<3.
c. If x=3, then 1<2.
d. If x=3, then 1>2.
7. Suppose someone says to you that the following statement is true: "If Jack is younger than his father, then Jack will not lose the contest." Did Jack win the contest?
8. If you are trying to prove that "A implies B" and you know B is false, do you want to show A is true or false? Explain.
9. By considering what happens when A is true and when A is false, it was decided that when trying to prove the statement "A implies B" is true, you can assume that A is true and your goal is to show that B is true. Use the same type of reasoning to derive another approach for proving that "A implies B" is true by considering what happens when B is true and when B is false.
Ch 7, problems 2,7,8.
2. Rewrite each of the following using nested quantifiers.
a. A set S of real numbers has the property that, no matter which element is chosen in the set, you can find another element in the set that is strictly larger.
b. A function of a single variable has the property that for some real number, the absolute
value of the function is always less than that real number
7. Prove that for every real number x>2, there is a real number y<0 such that x=2y/(1+y).
8. Prove that if S = {real numbers x>0 : x2 < 2}, then for every real number epsilon > 0, there is an element x in S such that x2 < 2 - epsilon.
Read Chapter 1 and the other problems within.
(turn in this one too)
II. For p a person and t a time, let fooled(p,t) be true if p is fooled at time t.
Use quantifier notation to write out "You can fool some of the people all of the time, and all of the people some of the time, but you can't fool all the people all of the time." (Abraham Lincoln, paraphrased)
More generally, how is "but" rendered in mathematics-ese?
posted by Allen Knutson @ 2:24 PM
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