For each of the following sets, prove that it is countable by showing that there is a bijection to that set from N.
For each of the following sets, prove that it is countable by showing that there is...
2. Determine whether the given sets are countable or uncountable. Justify each answer with a bijection (or table like we did with Q+) or using results from class/textbook. (a) {0, 1, 2} * N (b) A = {(x, y) : x2 + y2 = 1} (c) {0, 1} R Che set of all 2-element subsets of N (e) Real numbers with decimal representations consists of all 1s. (f) The set of all functions from {0,1} to N
Problem 22: Which of the following sets are countable? 1. N × Z 2. Q x Q x Q 3. R x R 4.(pe N p prime 7. Set of all infinite sequences of zeroes and ones. Problem 22: Which of the following sets are countable? 1. N × Z 2. Q x Q x Q 3. R x R 4.(pe N p prime 7. Set of all infinite sequences of zeroes and ones.
Question 7 Classify each of the following sets as finite, countable infinite, or uncountable (no proof is necessary): A=0 B = {2 ER: 0 < x < 0.0001} C=0 D=N E = {R} F= {n EN:n <9000} G=Z/5Z H = P(N) I= {n €Z:n > 50 J=Z Bonus: Give an example of a set with larger cardinality then any of the above sets.
Problem 8. Given each pair of sets, come up with a formula for a bijection between them You do not need to prove your function is a bijection. Your formula should not be complicated by any means 1. From (0, 1) to (211, 2019) 2. From [0, 1) to (0, 1] 3. From NU (o) to N. 4. From the set of even numbers to 2 5. From the set of odd numbers to Z. 6. r2'2 7. From R...
For each of the following pairs of sets, prove that they are equinumerous. Remember that we have two ways to do this: we can find a bijection explicitly, or we can prove that there is an injection in each direction and then use the Schr¨oder-Bernstein theorem. 4. N and Qd for d > 1 5. R and R x R {a + bi |2 =-1, a,bE R} is the complex numbers) 6. R and C (where C
In each of the following problems, three sets are described. One of the sets is not the same as the other two. In each case, find the set that is not like the others. (you do not need to prove, a very brief explanation is sufficient) (a) A = {a + b | a ∈ N, b ∈ N}; B = {a − b | a ∈ N, b ∈ N}; C = N. (b) A = {x^2 | x...
PROJECT 6.2 In this project you will construct an increasing function that is discontinuous at each rational point in (0, 1) and continuous at each irrational point in (0, 1). We will need two basic facts: a. The rational numbers in the interval (0, 1) can be arranged in a sequence rThis is true because the set of rational numbers is countable. (See Example 0.12 and Corollary 0.15.) b. Any rearrangement of an absolutely convergent series converges, and any sub-...
Sets, Please respond ASAP, Thank you 2) Recall another notation for the natural numbers, N, is Z+. We similarly define the negative integers by: 2. Too, for any set A and a e R, define: and Let B={x: x E Z+ & x is odd } (Recall a number I is said to be odd if 2k +1 for some k e z) Assume Z is our underlying background set for this problem. (a) Write an expression for 3 +...
Problem 5. Letf: Z+Zbyn -n. Let D, E S Z denote the sets of odd and even integers, respectively. (a) Prove that fD CE, where D denotes the image of D under f. (b) Is it true that D = E? Prove or disprove. (c) Describe the set f[El. Problem 6. Letf: R R be the function defined by fx) = x2 + 2x + 1. (a) Prove that f is not injective. Find all pairs of real numbers T1,...
#7. TRUE/FALSE. Determine the truth value of each sentence (no explanation required). ________(a) k in Z k2 + 9 = 0. ________(b) m, n in N, 5m 2n is in N. ________(c) x in R, if |x − 2| < 3, then |x| < 5. #8. For each statement, (i) write the statement in logical form with appropriate variables and quantifiers, (ii) write the negation in logical form, and (iii) write the negation in a clearly worded unambiguous English sentence....