Are all sets countable?
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.
Is the set of all finite languages countable?
The list of finite languages over a finite alphabet is countable.
What is not a countable set?
In mathematics, an uncountable set (or uncountably infinite set) is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal number is larger than that of the set of all natural numbers.
Are whole numbers countable?
So the assumption made before conducting the diagonalization is that the set of all whole numbers is countable/listable.
Can sets be infinite?
An infinite set is one that has no last element. An infinite set is a set that can be placed into a one-to-one correspondence with a proper subset of itself. A 1-1 correspondence between two sets A and B is a rule that associates each element of set A with one and only one element of set B and vice versa.
Which of the following are countable sets?
The sets N, Z, the set of all odd natural numbers, and the set of all even natural numbers are examples of sets that are countable and countably infinite.
Why is the set of languages uncountable?
Notice that while each string in a language must have finite length, the language itself may have infinitely many strings as illustrated in the second example just given. Our proof will be based on the fact that a contradiction is obtained if L{0,1} is countable.
Can a language be uncountable?
Language is always uncountable as a general concept or phenomenon.
Which of the following sets is are countable?
What is the difference between countable and uncountable set?
A set A is countably infinite if its cardinality is equal to the cardinality of the natural numbers N. A set is uncountable if it is infinite and not countably infinite.
Is the set RQ countable?
The set R of real numbers, which is uncountable, is the disjoint union of A=the set of irrationals and B=the set of rationals. The set B is countable. So if A was countable then R would be countable; but R is not countable so this is a contradiction. Hence A cannot be countable.
Why is the set of all programs countable?
The set of all programs is countable because every program can be written as a finite string over a finite alphabet. This is in fact the easiest way of proving the existence of non-computable reals: for every computable real, there is a program that computes it, and distinct reals are of necessity computed by distinct programs.
How do you prove a set of strings is countable?
For any finite set $X$, the set $X^*$ of all finite length strings over $X$ is countable (by the same sort of argument you would use to show the rationals are countable). Share Cite Follow answered Feb 20 ’13 at 23:07
Is it wrong to assert that $a_k$ is countable?
There is no problem with that union. However, it is wrong to assert that $A_k$is countable “because it is the union of various countable sets”. An arbitray union of countable sets doesn’t have to be countable. You have to justify that you have a countableunion of countable sets here.
How do you prove that $s$ is countably infinite?
If $S$is finite then the number of finite subsets of $S$is also finite, so in this case it’s easy to show that this is countable as well. In the case that $S$is countably infinite: