Consider the following 0-1 sequences and prove that they have the cardinality C. a) One sided seq...
Consider the following 0-1 sequences and prove that they have the cardinality C.
a) One sided sequences {01000111011010101...}
b) Two sided sequences {... 011010101010110...}
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Consider the following 0-1 sequences and prove that they have the cardinality C. a) One sided seq...
(a) Prove directly that the cardinality of the closed interval [0, 1] is equal to the...
(a) Prove directly that the cardinality of the closed interval [0, 1] is equal to the cardinality of the open interval (0, 1) by constructing a function f : [0, 1] → (0, 1) that is one-to-one and onto. (b) More generally, show that if S is an infinite set and {a,b} C S, then [S] = |S \ {a,b}\. (The notation S \ {a,b} is used to denote the set of all s in S such that s is...
Let S be a finite set with cardinality n>0. a. Prove, by constructing a bijection, that...
Let S be a finite set with cardinality n>0. a. Prove, by constructing a bijection, that the number of subsets of S of size k is equal to the number of subsets of size n- k. Be sure to prove that vour mapping is both injective and surjective. b. Prove, by constructing a bijection, that the number of odd-cardinality subsets of S is equal to the number of even-cardinality subsets of S. Be sure to prove that your mapping is...
4. As we have seen, sometimes two sets can have the same cardinality even when one...
4. As we have seen, sometimes two sets can have the same cardinality even when one seems obviously much bigger than the other. Show that the following sets have the same cardinality. In part a, give a complete proof by finding a bijection. In part b, consider our proof that the rationals are countable. (a) The interval (0,1) and the real numbers, R (b) The integers, Z, and the Cartesian Product of the integers with itself, Zx Z
(a) Recall that two sets have the same cardinality if there is a bijection between them...
(a) Recall that two sets have the same cardinality if there is a bijection between them and that Z is the set of all integers. Give an example of a bijection f: Z+Z which is different from the identity function. (b) For the following sets A prove that A has the same cardinality as the positive integers Z+ i. A= {r eZ+By Z r = y²} ii. A=Z 1.
Let ao 2 bo > 0, and consider the sequences an and bn defined by an...
Let ao 2 bo > 0, and consider the sequences an and bn defined by an + bn n20 (1) Compute an+l-bn+1 1n terms of Van-v/bn. (2) Prove that the sequence an is nonincreasing, that the sequence bn Is nonde- creasing, and that an 2 bn for all n 20 (3) Prove that VanVbn S Cr for all n20, where C> 0 and y>1 (give values of C and γ for which this inequality holds). Conclude that an-bn C,γ-n, where...
.6.29. Show that the following pairs of sets have the same cardinality. a) Integers divisible by ...
.6.29. Show that the following pairs of sets have the same cardinality. a) Integers divisible by 3, and the even positive integers (b) R, and the interval (0, oo). (c) The interval [0,2), and the set [5,6)U7,8) (d) The intervals (-oo,-1) and (-1,0)
.6.29. Show that the following pairs of sets have the same cardinality. a) Integers divisible by 3, and the even positive integers (b) R, and the interval (0, oo). (c) The interval [0,2), and the set [5,6)U7,8)...
2. (15 marks) Consider the abstract datatype SEQ whose objects are sequences of elements and whic...
do (b) please
2. (15 marks) Consider the abstract datatype SEQ whose objects are sequences of elements and which supports two operations . PREPEND(x, S), which inserts element r at the beginning of the sequence S and . ACCESS(S, i), which returns the ith element in the sequence Suppose that we represent S by a singly linked list. Then PREPEND(, S) takes 1 step and ACCESS(S, i) takes i steps, provided S has at least i elements Suppose that S...
Provide an ? N proof to prove that the following sequences converge. Question (e), please. 5....
Provide an ? N proof to prove that the following sequences
converge.
Question (e), please.
5. Provide an e – N proof to prove that the following sequences converge. (a) {ne cos(n)} (b) {zo Bom} (c) {(-1)In (n)} (d) an = 2 + 1 (@) an = V1 -
8.2 Determine the limits of the following sequences, and then prove your claims. (a) (n =...
8.2 Determine the limits of the following sequences, and then prove your claims. (a) (n = (b) b = 1 (c) C = Amis = 1 sinn
Consider the experiment of rolling six 6-sided dice. The sample space S is all length-6 sequences...
Consider the experiment of rolling six 6-sided dice. The sample space S is all length-6 sequences made up of integers 1 to 6, with replacement. (a) Find the probability of all dice yielding the same number. (b) Find the probability that all the numbers are distinct.
(a) Prove directly that the cardinality of the closed interval [0, 1] is equal to the cardinality of the open interval (0, 1) by constructing a function f : [0, 1] → (0, 1) that is one-to-one and onto. (b) More generally, show that if S is an infinite set and {a,b} C S, then [S] = |S \ {a,b}\. (The notation S \ {a,b} is used to denote the set of all s in S such that s is...
Let S be a finite set with cardinality n>0. a. Prove, by constructing a bijection, that the number of subsets of S of size k is equal to the number of subsets of size n- k. Be sure to prove that vour mapping is both injective and surjective. b. Prove, by constructing a bijection, that the number of odd-cardinality subsets of S is equal to the number of even-cardinality subsets of S. Be sure to prove that your mapping is...
4. As we have seen, sometimes two sets can have the same cardinality even when one seems obviously much bigger than the other. Show that the following sets have the same cardinality. In part a, give a complete proof by finding a bijection. In part b, consider our proof that the rationals are countable. (a) The interval (0,1) and the real numbers, R (b) The integers, Z, and the Cartesian Product of the integers with itself, Zx Z
(a) Recall that two sets have the same cardinality if there is a bijection between them and that Z is the set of all integers. Give an example of a bijection f: Z+Z which is different from the identity function. (b) For the following sets A prove that A has the same cardinality as the positive integers Z+ i. A= {r eZ+By Z r = y²} ii. A=Z 1.
Let ao 2 bo > 0, and consider the sequences an and bn defined by an + bn n20 (1) Compute an+l-bn+1 1n terms of Van-v/bn. (2) Prove that the sequence an is nonincreasing, that the sequence bn Is nonde- creasing, and that an 2 bn for all n 20 (3) Prove that VanVbn S Cr for all n20, where C> 0 and y>1 (give values of C and γ for which this inequality holds). Conclude that an-bn C,γ-n, where...
.6.29. Show that the following pairs of sets have the same cardinality. a) Integers divisible by 3, and the even positive integers (b) R, and the interval (0, oo). (c) The interval [0,2), and the set [5,6)U7,8) (d) The intervals (-oo,-1) and (-1,0)
.6.29. Show that the following pairs of sets have the same cardinality. a) Integers divisible by 3, and the even positive integers (b) R, and the interval (0, oo). (c) The interval [0,2), and the set [5,6)U7,8)...
do (b) please
2. (15 marks) Consider the abstract datatype SEQ whose objects are sequences of elements and which supports two operations . PREPEND(x, S), which inserts element r at the beginning of the sequence S and . ACCESS(S, i), which returns the ith element in the sequence Suppose that we represent S by a singly linked list. Then PREPEND(, S) takes 1 step and ACCESS(S, i) takes i steps, provided S has at least i elements Suppose that S...
Provide an ? N proof to prove that the following sequences
converge.
Question (e), please.
5. Provide an e – N proof to prove that the following sequences converge. (a) {ne cos(n)} (b) {zo Bom} (c) {(-1)In (n)} (d) an = 2 + 1 (@) an = V1 -
8.2 Determine the limits of the following sequences, and then prove your claims. (a) (n = (b) b = 1 (c) C = Amis = 1 sinn
Consider the experiment of rolling six 6-sided dice. The sample space S is all length-6 sequences made up of integers 1 to 6, with replacement. (a) Find the probability of all dice yielding the same number. (b) Find the probability that all the numbers are distinct.
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