* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download HW-06 due 02/22
Survey
Document related concepts
Abuse of notation wikipedia , lookup
Law of large numbers wikipedia , lookup
Location arithmetic wikipedia , lookup
Ethnomathematics wikipedia , lookup
Infinitesimal wikipedia , lookup
History of the function concept wikipedia , lookup
Georg Cantor's first set theory article wikipedia , lookup
Foundations of mathematics wikipedia , lookup
Surreal number wikipedia , lookup
Fundamental theorem of algebra wikipedia , lookup
Large numbers wikipedia , lookup
Mathematics of radio engineering wikipedia , lookup
Real number wikipedia , lookup
Non-standard calculus wikipedia , lookup
Transcript
CmSc180 Discrete Mathematics Homework 06 due 02/22 1. Determine whether the following arguments are valid or invalid. a. Sandra knows COBOL and Sandra knows C. Therefore Sandra knows C b. If the number is larger than 2, then its square is larger than 4. This number is not larger than 2. Therefore the square of this number is not larger than 4. c. If at least one of these two numbers is divisible by 6, then the product is divisible by 3. Neither of these two numbers is divisible by 6. Therefore the product is not divisible by 3. d. If there are as many rational numbers as there are irrational numbers, then the set of all rational numbers is infinite. The set of all rational numbers is infinite. Therefore there are as many rational numbers as there are irrational numbers. 2. Suppose you are visiting an island where two kinds of people live: knights who always tell the truth, and knaves who always lie. Two natives A and B approach you, and B says: "Both of us are knaves". What are A and B? Explain your reasoning. 3. Prove by mathematical induction 1 + 2 + 22 + 23 + … + 2n = 2(n+1) - 1, n = 0, 1, 2, 3, … 4. Let the universal set be the set R of all real numbers and let A = {x R | -2 x 3}, B = { x R | -1 < x < 4}. Find each of the following: AB = AB = ~A = 1 ~B = ~A ~B = ~A ~B = ~(A B) = ~(A B) = 5. Prove using element proof that (A - B) - C = A - (B C) 6. Find the powerset of 6.1. A = {{a}, {b}} 6.2. A = 6.3. A = {} 7. Let the universal set be all people. Within the universal set, A is the set of all computer programmers, B is the set of all accountants, C is the set of all women, D is the set of all people of age 40 and older. What are the following sets: a. ~(A B) b. (A B) ~C c. ~D C A 8. Let A = {1,2,3,4,5}, B = {a, b, c, d, e} a. Give a one-to-one function from A to B. Is it a bijection? b. Give a function from A to B which is not one-to-one. Is it an onto function? c. Give a function from A to B which is not onto. Is it one-to-one? 9. Let A = {a,b,c}, B = {1,2} a. Give an onto function from A to B. Is it one-to-one? b. Give a function from B to A which is one-to-one. Is it “onto”? 2