“HILBERT’S GRAND HOTEL: We now describe a paradox that shows that something impossible with finite sets may be possible with infinite sets. The famous mathematician David Hilbert invented the notion of the Grand Hotel, which has a countably infinite number of rooms, each occupied by a guest. When a new guest arrives at a hotel […]
“HILBERT’S GRAND HOTEL: We now describe a paradox that shows that something impossible with finite sets may be possible with infinite sets. The famous mathematician David Hilbert invented the notion of the Grand Hotel, which has a countably infinite number of rooms, each occupied by a guest. When a new guest arrives at a hotel
Relevant definitions: “A set that is either finite or has the same cardinality as the set of positive integers is called countable. A set that is not countable is called uncountable. When an infinite set S is countable, we denote the cardinality of S by א0 (where א is aleph, the first letter of the
Relevant definitions: “A set that is either finite or has the same cardinality as the set of positive integers is called countable. A set that is not countable is called uncountable. When an infinite set S is countable, we denote the cardinality of S by א0 (where א is aleph, the first letter of the
Relevant definitions: “A set that is either finite or has the same cardinality as the set of positive integers is called countable. A set that is not countable is called uncountable. When an infinite set S is countable, we denote the cardinality of S by א0 (where א is aleph, the first letter of the
As we always do, let’s first start with the relevant definitions we need to solve this exercise. “A set that is either finite or has the same cardinality as the set of positive integers is called countable. A set that is not countable is called uncountable. When an infinite set S is countable, we denote
Let’s refresh the definitions we will need to solve this exercise. Definition: “Let A and B be sets. The difference of A and B, denoted by A − B, is the set containing those elements that are in A but not in B. The difference of A and B is also called the complement of
Give an example of two uncountable sets A and B such that A−B is Read More »
Let’s start with the definitions. Definition: “Let S be a set. If there are exactly n distinct elements in S where n is a nonnegative integer, we say that S is a finite set and that n is the cardinality of S. The cardinality of S is denoted by |S|.” Definition: “If there is a
Show that if A and B are sets and A⊂B then |A|≤|B| Read More »
As usual, let’s review the definitions before starting to solve the exercise. Definition: “The sets A and B have the same cardinality if and only if there is a one-to-one correspondence from A to B. When A and B have the same cardinality, we write |A| = |B|.” Definition: “A set that is either finite
In this case, we first must revisit the paradox of Hilbert’s Hotel. “HILBERT’S GRAND HOTEL: We now describe a paradox that shows that something impossible with finite sets may be possible with infinite sets. The famous mathematician David Hilbert invented the notion of the Grand Hotel, which has a countably infinite number of rooms, each
