Question

1.

Let A and B be sets. The set B is of at least the same size as the set A if and only if (mark all correct answers)

there is a bijection from A to B |

there is a one-to-one function from A to B |

there is a one-to-one function from B to A |

there is an onto function from B to A |

A is a proper subset of B |

2.

Which of these sets are countable? (mark all correct answers)

the set of all 3-tuples (a,b,c), where a, b, and c are integers |

the set of all real numbers |

the set of all infinite sequences over natural numbers |

the set of irrational numbers |

the set of all functions from R to {0,1} |

the set of all binary strings of length 113 |

Answer #1

Implement function subset that takes a set of
positive integers and returns asubset of it that includes only
those numbers that can form a sequence. It's ok if more than one
sequences co-exist in the subset. You may not use lists, tuples,
strings, dictionaries; only set functions/methods are allowed.
Examples:
subset({0,4,11,5,3,2,7,9}) ->
{4,5,3,2}
subset({3,1,6,8,2,12,9}) ->
{3,1,2,8,9}
True
False
In Python Please

1. [10 marks] We begin with some mathematics regarding
uncountability. Let N = {0, 1, 2, 3, . . .} denote the set of
natural numbers.
(a) [5 marks] Prove that the set of binary numbers has the same
size as N by giving a bijection between the binary numbers and
N.
(b) [5 marks] Let B denote the set of all infinite sequences
over the English alphabet. Show that B is uncountable using a proof
by diagonalization.

(a) Let A and B be countably infinite sets. Decide whether the
following are true for all, some (but not all), or no such sets,
and give reasons for your answers. A ∪B is countably infinite A
∩B is countably infinite A\B is countably infinite, where A ∖ B =
{ x | x ∈ A ∧ X ∉ B }. (b) Let F be the set of all total unary
functions f : N → N...

Let S denote the set of all possible finite binary strings, i.e.
strings of finite length made up of only 0s and 1s, and no other
characters. E.g., 010100100001 is a finite binary string but
100ff101 is not because it contains characters other than 0, 1.
a. Give an informal proof arguing why this set should be
countable. Even though the language of your proof can be informal,
it must clearly explain the reasons why you think the set should...

Problem 3 Countable and Uncountable Sets
(a) Show that there are uncountably infinite many real numbers
in the interval (0, 1). (Hint: Prove this by contradiction.
Specifically, (i) assume that there are countably infinite real
numbers in (0, 1) and denote them as x1, x2, x3, · · · ; (ii)
express each real number x1 between 0 and 1 in decimal expansion;
(iii) construct a number y whose digits are either 1 or 2. Can you
find a way...

For each of the following sets X and collections T of open
subsets decide whether the pair X, T satisfies the axioms of a
topological space. If it does, determine the connected components
of X. If it is not a topological space then exhibit one axiom that
fails.
(a) X = {1, 2, 3, 4} and T = {∅, {1}, {1, 2}, {2, 3}, {1, 2, 3},
{1, 2, 3, 4}}.
(b) X = {1, 2, 3, 4} and T...

.Unless otherwise noted, all sets in this module are finite.
Prove the following statements!
1. There is a bijection from the positive odd numbers to the
integers divisible by 3.
2. There is an injection f : Q→N.
3. If f : N→R is a function, then it is not surjective.

1. Consider the following situation: The universal set U
is given by: U = {?|? ? ? , ? ≤ 12}
A is a proper subset of U, with those numbers that are
divisible by 4.
B is a proper subset of U, with those numbers that are
divisible by 3.
C is a proper subset of U, with those numbers that are
divisible by 2
a) Using Roster Notation, list the elements of sets U, A, B and
C....

15.)
a) Show that the real numbers between 0 and 1 have the same
cardinality as the real numbers between 0 and pi/2. (Hint: Find a
simple bijection from one set to the other.)
b) Show that the real numbers between 0 and pi/2 have the same
cardinality as all nonnegative real numbers. (Hint: What is a
function whose graph goes from 0 to positive infinity as x goes
from 0 to pi/2?)
c) Use parts a and b to...

Answer the following brief question:
(1) Given a set X the power set P(X) is ...
(2) Let X, Y be two infinite sets. Suppose there exists an
injective map f : X → Y but no surjective map X → Y . What can one
say about the cardinalities card(X) and card(Y ) ?
(3) How many subsets of cardinality 7 are there in a set of
cardinality 10 ?
(4) How many functions are there from X =...

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