Theorem: The set of all finite subsets of the natural numbers is countable. The elements of any finite subset can be ordered into a finite sequence. There are only countably many finite sequences, so there are only countably many finite subsets.
Can a finite set be countable?
Every subset of a finite set is finite. The set of values of a function when applied to the elements of a finite set is finite. All finite sets are countable, but not all countable sets are finite. (However, some authors use countable to mean countable infinite, so don’t consider finite sets to be countable.)
Can a set be infinite?
An infinite set is a set whose elements cannot be counted. An infinite set is a set that has no last element. An infinite set is a set that can be brought into one-to-one correspondence with a proper subset of itself. … A set is infinite if it can be mapped to a proper subset.
What does it mean that a set is countable?
Countable means that there is a bijection between the given set and the set N. Indeed, as you point out, this leads to subtleties in some of the proofs.
Are countable and countable the same?
The difference between countable and countable When used as an adjective, countable means that it is possible to be counted while countable means that the natural numbers can be assigned a bijection.
What is the difference between a finite set and a countable set?
A countable set is either a finite set or an infinitely countable set. Whether finite or infinite, the elements of a countable set can always be individually counted, and although counting may never end, each element of the set is associated with a unique natural number.
Is the empty set finite?
The empty set is also considered a finite set and its cardinal number is 0.
How do you know if a set is infinite?
An infinite set is endless from start or end, but both sides can have continuity, unlike the finite set where there are both start and end elements. If a set has infinitely many elements, then it is infinite, and if the elements are countable, then it is finite. 18
Can an infinite set have a finite subset?
Although we haven’t defined the terms yet, we’ll see that one thing that distinguishes an infinite set from a finite set is that an infinite set can be equivalent to each of its own subsets, while a finite set can’t be equivalent can each have its own subsets.
How do you prove Denumerable?
Theorem: If A is countable and x ∉ A, then A ∪ {x} is countable.
What are examples of countable sets?
Examples of countable sets are integers, algebraic numbers, and rational numbers. Georg Cantor showed that the number of real numbers is rigorously greater than an infinite countable set, and the postulate that this number, the so-called continuum, is equal to aleph1 is called the continuum hypothesis.
Is countable countable infinite?
countable if finite or countable. Countable sets are sometimes referred to as countably infinite. 21